Step No. 1: Seek out lotteries with positive expected value.
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If the sole criterion were expected value, you should play the game.
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"The long term expected value of your ETNs is zero," the prospectus says.
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But development teams must take calculated risks where the expected value is positive.
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But watch what happens when he encounters a lottery with positive expected value.
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Ordinary citizens, in broad daylight, could be heard talking combinatorics and expected value.
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It did not disclose the expected value of the investment or Sanpo's financial results.
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Then the expected value of blocking Garland drops to 3.5 — worse than confirming him.
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Each bet thus had (what an economist would call) the same expected value: $7.50.
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He did not name the companies or give the expected value of the IPOs.
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Repeat this enough times, and the ticket's expected value may rise above its price.
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But of course, no one plays the lottery for the expected value of a ticket.
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In sum, the expected value of solving the AI control problem could be astronomically high.
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You can also use AV to measure and compare the expected value of draft picks.
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Maybe it's low, but the expected value of having an escape hatch is pretty high.
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Only 7% of respondents expected value stocks to outperform growth over the next 12 months.
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Futures capturing the expected value of the S&P 500 stockmarket index sank by around 0.9%.
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But so does the chance of multiple winners, which lowers the expected value of a ticket.
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It's a bet with clearly negative expected value, except it's the American people who are holding the downside.
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He asks if the lottery prize is high enough to be a net gain in expected value terms.
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In addition, we've modeled our pricing based off expected value each robotic kitchen assistant can provide at scale.
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That's because the payout you can anticipate from buying a ticket (the "expected value") isn't worth the investment.
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Even the estimated cash value of the jackpot — around $930 million — generates an expected value north of $3.50.
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Net bids were $22019 billion less than the Congressional Budget Office's expected value of $25 billion, and the $.
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With only 7 million possible number combinations, that gave each $1 ticket an expected value of nearly $4.
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"Expected value" distills the multifaceted lottery ticket, with all its prizes and probabilities, down to a one-number summary.
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So to those enamored with expected value, the latter is a golden opportunity, while the former is fool's gold.
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Mathematician and Wharton Ph.D. candidate Seth Neel ran an independent analysis for the impact on expected value across different ticket amounts with the jackpot at $1.5 billion, and even after factoring in taxes, lump sum conversions, and lesser prizes: Of course, no one plays the lottery for the expected value of a ticket.
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Split it three ways and your expected value drops below $1.40, a return on investment of about negative 30 percent.
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Thus, it shares the same expected value as A. Yet if you're like most people, you've got a strong preference.
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That's how we got this week's Mega Millions blockbuster, where each $2 ticket had an expected value of over $5.
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For Mega Millions, the cost is constant at $1 per ticket, so the focus falls squarely on a ticket's expected value.
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If this so-called "expected value" of the policy is less than the premium, the insurance "investment" fails the break-even test.
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This process gives the expected value of the lottery ticket, and is a useful tool in analyzing gambling games like a lottery.
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Calculating the expected value, you start to salivate: $1 billion spent on tickets would yield an expected return of $10 billion. Irresistible!
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Expected value is a long-run average, and with Gates's offer, you'll exhaust your finances well before the "long run" ever arrives.
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He can stay in front of anyone on defense, at least long enough to put his finger on the scale of expected value.
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He had been blamed by investors for the postponement of WeWork's IPO, which was shelved after a sharp drop in its expected value.
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No terms were announced, but the expected value based on Brown's No. 25 draft slot is $11.8 million over the first four years.
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"They've reduced the expected value of sending a message to essentially zero whereas it used to be the highest in the industry," Dean explained.
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First off, it's worth defining the measure that gamblers and statisticians look to in deciding whether a bet is worth the wager: expected value.
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At that level, the expected value of a ticket comes to 64 cents after accounting for taxes, net present value and the lesser prizes.
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But teams including the Cowboys, Patriots and 49ers have higher valuations than would be expected, while the Panthers come in below their expected value.
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Farmers have the option of different levels of insurance coverage for their crops, ranging from 220006 percent to 2202 percent of the expected value.
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In the case of Powerball, the cost to play is constant at $2 per ticket, so the focus falls squarely on a ticket's expected value.
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Last week, our picks included running back Wayne Gallman and tight end Will Dissly, who both far outperformed their expected value at their listed prices.
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Last week, our picks included wide receiver Nelson Agholor and running back Frank Gore, who both far outperformed their expected value at their listed prices.
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Last week, our picks included tight end Mark Andrews and wide receiver DK Metcalf, who both far outperformed their expected value at their listed prices.
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It's the Educated Fool, a rare creature who does with "expected value" what the foolish always do with education: mistake partial truth for total wisdom.
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Perhaps the ultimate repudiation of expected value is the abstract possibility of $1 tickets like this: If you buy 10 tickets, you're likely to win $1.
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The researchers ran an algorithm that relied on 11 of the qubits, called the Bernstein-Vazirani algorithm, which returned the expected value 73 percent of the time.
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But dividing the jackpot cash in two, the expected value of your ticket drops to $1.91, less than the $123 you shelled out in the first place.
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Only very rarely have so many players piled in that the expected value of the jackpot has fallen, and mostly when there are huge jackpots on offer.
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We have two costs to compute: the utility loss from waiting at the airport, and the expected value of the utility loss from maybe missing your flight.
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With no winner, the jackpot creeps closer to the $400 million threshold -- where the expected value exceeds the cost of the ticket, according to Los Angeles Times.
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Doing so wouldn't make much sense seeing as the $2 cost of a Powerball ticket is almost always more than the expected value (the house essentially always wins).
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So maybe the equity is smaller and more expensive, but ultimately, if the startup is more likely to be successful, the expected value function might actually be favorable. Maybe.
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For example, if the jackpot is valued at $100 million, the expected value of a ticket is just 66 cents, far less than the $1.53 you'd be shelling out.
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The drawback is that calculations of the magnitude of this "vacuum energy" give a figure at least 10{+6}{+0} times greater than the expected value of the cosmological constant.
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Doing so wouldn't make much sense seeing as the $1 cost of a Mega Millions ticket is designed to be more than the expected value (the house essentially always wins).
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This week, with a $1.5 billion jackpot on the line, the expected value is increased to more than $5 (including the nonjackpot prizes), making a $2 ticket a good investment.
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Giles has enough talent and upside to exceed the expected value of a typical 20th overall pick, and there's the glimmer of a possibility that he could be a star.
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"The thing we focus on is expected value," said Ms. Tuna, a former journalist who runs their foundation while her husband focuses on running his current tech start-up, Asana.
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Does that mean that that even with your philanthropy or advocacy you take on greater risks that are a long shot at achieving, but perhaps have a high-expected value return?
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That in turn mathematically lowers the expected value of an individual lottery ticket, explaining why a larger jackpot that stirs up intense news coverage might not always be the best proposition.
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Despite that eye-popping headline prize, an analysis factoring the possible impact of taxes suggested that the expected value of a ticket would be negative, making the ticket a poor investment.
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One can reasonably expect that these funds will be aggressive in searching for opportunities to identify patents for assertion and work out financing deals based on the expected value of those patents.
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Well, that's a question for our next player... At first glance, this character looks an awful lot like the Educated Fool: the same scheming grin, the same obsessive focus on expected value.
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I do some activity as a donor, I do some activity as a citizen, and I hope the things I advocate as the latter, possibly with a higher expected value, actually come to fruition.
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"When we take into account the financial impacts of efforts to cut emissions, we still find the expected value of financial assets is higher in a world that limits warming to 2°C," Dietz said.
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The bigger loss related to a write-down in the value of its General Medicine unit by $1.09 billion, due to delays in clinical studies and a reduction in the expected value of some R&D projects.
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Still, however unlikely big outcomes are, the possibility of being a part of the next Facebook or Uber is tempting, and taking a job at a brand new company may even rational on an expected value basis.
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Running a regression on Mega Millions drawings reveals the sweet spot for maximizing the expected value of a $1 ticket appears to be around a jackpot of $547 million, or just $7 million more than Friday's advertised jackpot.
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So if we only catch 50 percent of thieves, and account for my expected value of a $10,000 jewel, the actual [deterrent] value is $20,000 — because there's only a 50 percent chance I'll have to pay it back.
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In 2016, the Open Philanthropy Project published "Hits-Based Giving," an argument that philanthropists should accept more risk because many of the giving opportunities with the highest expected value would be ones that were unlikely to pan out.
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He was finishing his first season as a Lindblad expedition leader, was clearly exhausted, and was under intense pressure to deliver the trip of a lifetime to customers who, not being plutocrats after all, expected value for their money.
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As long as the winner's curse ensured that at least one team would always pay an ageing star far more than his expected value, this arrangement worked as a de facto cross-subsidy from younger union members to older ones.
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Austria's Do&Co and Switzerland's Gategroup are expected to make offers for the European LSG operations, the people said, adding that given its low profitability and low expected value even medium-sized Do&Co could do a deal without a partner.
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The government, through the Centers for Medicare & Medicaid Services (CMS) or alternatively state agencies, would also charge a risk-based premium per qualifying individual, which, together with an expected value of deductibles, would be capped at a maximum percentage of income.
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Before you're ready to purchase your first home, you should sock away at least 20% of your home's expected value to use as a down payment, Orman tells CNBC Make It. Next, make sure you build up an eight-month emergency fund.
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Self-selection: Most founders are smart, driven and skilled people whose résumé could almost certainly land them a job with a higher lifetime expected value (the median salary at Facebook is now $240,000), but they still choose the grueling, uncertain and more creative founder journey.
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"There's unquestionably a market-signaling effect in raising money at a valuation that, in our opinion, doesn't come close to the ultimate expected value of the company," said Liu, in conversation with senior Recode editor Teddy Schleifer at a recent industry event hosted by this editor.
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The expected value — calculated by multiplying the chance of success by the reward — reveals that the average extra point is worth 22000 points (22 percent times 22) over the past two seasons while the average 22015-point try — 22 percent — is worth 22 points (153 percent times 215).
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The answer, familiar to international macroeconomists, is that the dollar rises above its long-run expected value, so that people expect it to decline in the future – and the extent of the rise is determined by how high the dollar has to go so that expected depreciation outweighs the rise in after-tax returns compared with other countries.
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We go along with the present strategic plan, actually we are not putting it into question, not even of course the legal rights and the expected value that those who invested and continue to invest on TAP, we expect and, so this is in some sense and in a nutshell, we didn't change the nature of the company as a private company, it will continue to be so and the presence of the government is only restricted to the strategic review of the company.
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The expected value of a monetary gamble is a weighted average, in which each possible outcome is weighted by its probability of occurrence. The expected value of the gamble in this example is .85 X $1000 + .15 X $0 = $850, which exceeds the expected value of $800 associated with the sure thing.
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Warren Thorngate, a social psychologist, implemented ten simple decision rules or heuristics in a computer program. He determined how often each heuristic selected alternatives with highest-through-lowest expected value in a series of randomly-generated decision situations. He found that most of the simulated heuristics selected alternatives with highest expected value and almost never selected alternatives with lowest expected value.
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In the presence of risky outcomes, a human decision maker does not always choose the option with higher expected value investments. For example, suppose there is a choice between a guaranteed payment of $1.00, and a gamble in which the probability of getting a $100 payment is 1 in 80 and the alternative, far more likely outcome (79 out of 80) is receiving $0. The expected value of the first alternative is $1.00 and the expected value of the second alternative is $1.25. According to expected value theory, people should choose the $100-or-nothing gamble; however, as stressed by expected utility theory, some people are risk averse enough to prefer the sure thing, despite its lower expected value.
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Efficiency factor is a ratio of some measure of performance to an expected value.
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The expected value is then given by, E(X1) + E(X2) + ... + E(Xn). Since E(Xk) = P(Xk = 1) = 1/(n − k + 1), the sought expected value is 1/n + 1/(n − 1) + 1/(n − 2) + ... + 1 = Hn (the nth harmonic number).
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The expected value of U is 0. For large sample sizes U is distributed normally.
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Find the expected value of that result. This will be the approximation for the variance of z.
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Definition: Let \xi be an uncertain variable with finite expected value e. Then the variance of \xi is defined by :::V[\xi]=E[(\xi-e)^2]. Theorem: If \xi be an uncertain variable with finite expected value, a and b are real numbers, then :::V[a\xi+b]=a^2V[\xi].
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People with less risk aversion would choose the riskier, higher-expected-value gamble. This is precedence for utility theory.
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A mathematically correct solution involving sampling was offered by William Feller. In order to understand Feller's answer correctly, sufficient knowledge about probability theory and statistics is necessary, but it can be understood intuitively "to perform this game with a large number of people and calculate the expected value from the sample extraction". In this method, when the games of infinite number of times are possible, the expected value will be infinity, and in the case of finite, the expected value will be a much smaller value.
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In probability theory, a convex function applied to the expected value of a random variable is always bounded above by the expected value of the convex function of the random variable. This result, known as Jensen's inequality, can be used to deduce inequalities such as the arithmetic-geometric mean inequality and Hölder's inequality.
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If a distribution does not have a finite expected value, as is the case for the Cauchy distribution, then the variance cannot be finite either. However, some distributions may not have a finite variance, despite their expected value being finite. An example is a Pareto distribution whose index k satisfies 1 < k \leq 2.
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This suggests that loss attention may be more robust than loss aversion. Still, one might argue that loss aversion is more parsimonious than loss attention. Additional phenomena explained by loss attention: Increased expected value maximization with losses – It was found that individuals are more likely to select choice options with higher expected value (namely, mean outcome) in tasks where outcomes are framed as losses than when they are framed as gains. Yechiam and Hochman found that this effect occurred even when the alternative producing higher expected value was the one that included minor losses.
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A deviation that is a difference between an observed value and the true value of a quantity of interest (where true value denotes the Expected Value, such as the population mean) is an error. A deviation that is the difference between the observed value and an estimate of the true value (e.g. the sample mean; the Expected Value of a sample can be used as an estimate of the Expected Value of the population) is a residual. These concepts are applicable for data at the interval and ratio levels of measurement.
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If some of the probabilities \Pr\,(X=c_i) of an individual outcome c_i are unequal, then the expected value is defined to be the probability-weighted average of the c_is, that is, the sum of the n products c_i\cdot \Pr\,(X=c_i). The expected value of a general random variable involves integration in the sense of Lebesgue.
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VoI is sometimes distinguished into value of perfect information, also called value of clairvoyance (VoC), and value of imperfect information. They are closely related to the widely known expected value of perfect information and expected value of sample information. Note that VoI is not necessarily equal to "value of decision situation with perfect information" - "value of current decision situation" as commonly understood.
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See also main article: expected value. ; buck : Marker to indicate which player is dealer (or last to act). See button. ; bug : A limited wild card.
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When the total corrected sum of squares in an ANOVA is partitioned into several components, each attributed to the effect of a particular predictor variable, each of the sums of squares in that partition is a random variable that has an expected value. That expected value divided by the corresponding number of degrees of freedom is the expected mean square for that predictor variable.
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If a split is possible, the equity also includes the probability of winning a split times the size of that split. ; expectation, expected value, EV : See main article: expected value. Used in poker to mean profitability in the long run. ; exposed card : A card whose face has been deliberately or accidentally revealed to players normally not entitled to that information during the play of the game.
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If every object in a collection of objects fails to have a certain property, then the probability that a random object chosen from the collection has that property is zero. Similarly, showing that the probability is (strictly) less than 1 can be used to prove the existence of an object that does not satisfy the prescribed properties. Another way to use the probabilistic method is by calculating the expected value of some random variable. If it can be shown that the random variable can take on a value less than the expected value, this proves that the random variable can also take on some value greater than the expected value.
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The fundamental theorem of finance states that the price of assembling such a portfolio will be equal to its expected value under the appropriate risk-neutral measure.
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The expected force measures node influence from an epidemiological perspective. It is the expected value of the force of infection generated by the node after two transmissions.
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In this simulation the x data had a mean of 10 and a standard deviation of 2. Thus the naive expected value for z would of course be 100. The "biased mean" vertical line is found using the expression above for μz, and it agrees well with the observed mean (i.e., calculated from the data; dashed vertical line), and the biased mean is above the "expected" value of 100.
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In addition, there are keywords for the expected value, value at risk (VaR) and conditional value at risk (CVaR). Variables that are risk measures can feature in the objective equation or in constraints. EMP SP facilitates the optimization of a single risk measure or a combination of risk measures (for example, the weighted sum of Expected Value and CVaR). In addition, the modeler can choose to trade off risk measures.
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Both frequentist and Bayesian statistical theory involve making a decision based on the expected value of the loss function; however, this quantity is defined differently under the two paradigms.
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Compare to expected value analysis, whose conclusion is of the form: "this strategy yields E(X)=n." Minimax thus can be used on ordinal data, and can be more transparent.
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De Witt's approach was especially insightful and ahead of its time. In modern terminology, De Witt expressed the value of a life annuity as the expected value of a random variable.
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The researchers divided this distance by the speed of light in vacuum to predict what the neutrino travel time should be. They compared this expected value to the measured travel time.
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Expected value of sample information (EVSI) is a relaxation of the expected value of perfect information (EVPI) metric, which encodes the increase of utility that would be obtained if one were to learn the true underlying state, x. Essentially EVPI indicates the value of perfect information, while EVSI indicates the value of some limited and incomplete information. The expected value of including uncertainty (EVIU) compares the value of modeling uncertain information as compared to modeling a situation without taking uncertainty into account. Since the impact of uncertainty on computed results is often analysed using Monte Carlo methods, EVIU appears to be very similar to the value of carrying out an analysis using a Monte Carlo sample, which closely resembles in statement the notion captured with EVSI.
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We are interested in a bidder's expected value from the auction (the expected value of the item, minus the expected price) conditioned on the assumption that the bidder wins the auction. It turns out that for a bidder's true estimate the expected value is negative, meaning that on average the winning bidder is overpaying. Savvy bidders will avoid the winner's curse by bid shading, or placing a bid that is below their ex ante estimation of the value of the item for sale—but equal to their ex post belief about the value of the item, given that they win the auction. The key point is that winning the auction is bad news about the value of the item for the winner.
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In statistics, the bias (or bias function) of an estimator is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called unbiased. In statistics, "bias" is an objective property of an estimator. Bias can also be measured with respect to the median, rather than the mean (expected value), in which case one distinguishes median-unbiased from the usual mean-unbiasedness property.
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Note that to do this we cannot simply double the one-tailed p-value unless the probability of the event is 1/2. This is because the binomial distribution becomes asymmetric as that probability deviates from 1/2. There are two methods to define the two-tailed p-value. One method is to sum the probability that the total deviation in numbers of events in either direction from the expected value is either more than or less than the expected value.
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In general, although when one gamble first-order stochastically dominates a second gamble, the expected value of the payoff under the first will be greater than the expected value of the payoff under the second, the converse is not true: one cannot order lotteries with regard to stochastic dominance simply by comparing the means of their probability distributions. For instance, in the above example C has a higher mean (2) than does A (5/3), yet C does not first-order dominate A.
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Here the value of the option is calculated using the risk neutrality assumption. Under this assumption, the "expected value" (as opposed to "locked in" value) is discounted. The expected value is calculated using the intrinsic values from the later two nodes: "Option up" and "Option down", with u and d as price multipliers as above. These are then weighted by their respective probabilities: "probability" p of an up move in the underlying, and "probability" (1-p) of a down move.
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In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – given that a certain set of "conditions" is known to occur. If the random variable can take on only a finite number of values, the “conditions” are that the variable can only take on a subset of those values. More formally, in the case when the random variable is defined over a discrete probability space, the "conditions" are a partition of this probability space. With multiple random variables, for one random variable to be mean independent of all others—both individually and collectively—means that each conditional expectation equals the random variable's (unconditional) expected value.
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The term gambler's ruin is a statistical concept, most commonly expressed as the fact that a gambler playing a negative expected value game will eventually go broke, regardless of their betting system. The original meaning of the term is that a persistent gambler who raises his bet to a fixed fraction of bankroll when he wins, but does not reduce it when he loses, will eventually and inevitably go broke, even if he has a positive expected value on each bet. Another common meaning is that a persistent gambler with finite wealth, playing a fair game (that is, each bet has expected value zero to both sides) will eventually and inevitably go broke against an opponent with infinite wealth. Such a situation can be modeled by a random walk on the real number line.
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Its decision problem is to choose q so as to maximize the expected utility of profit: :Maximize Eu(pq – c(q) – g), where E is the expected value operator, u is the firm's utility function, c is its variable cost function, and g is its fixed cost. All possible distributions of the firm's random revenue pq, based on all possible choices of q, are location- scale related; so the decision problem can be framed in terms of the expected value and variance of revenue.
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Risk aversion is a preference for a sure outcome over a gamble with higher or equal expected value. Conversely, the rejection of a sure thing in favor of a gamble of lower or equal expected value is known as risk-seeking behavior. The psychophysics of chance induce overweighting of sure things and of improbable events, relative to events of moderate probability. Underweighting of moderate and high probabilities relative to sure things contributes to risk aversion in the realm of gains by reducing the attractiveness of positive gambles.
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The probability distribution of the number of fixed points in a uniformly distributed random permutation approaches a Poisson distribution with expected value 1 as n grows. In particular, it is an elegant application of the inclusion–exclusion principle to show that the probability that there are no fixed points approaches 1/e. When n is big enough, the probability distribution of fixed points is almost the Poisson distribution with expected value 1. The first n moments of this distribution are exactly those of the Poisson distribution.
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In the notation conventional among physicists, the Wick product is often denoted thus: :: X_1, \dots, X_k:\, and the angle-bracket notation :\langle X \rangle\, is used to denote the expected value of the random variable X.
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The area of choice under uncertainty represents the heart of decision theory. Known from the 17th century (Blaise Pascal invoked it in his famous wager, which is contained in his Pensées, published in 1670), the idea of expected value is that, when faced with a number of actions, each of which could give rise to more than one possible outcome with different probabilities, the rational procedure is to identify all possible outcomes, determine their values (positive or negative) and the probabilities that will result from each course of action, and multiply the two to give an "expected value", or the average expectation for an outcome; the action to be chosen should be the one that gives rise to the highest total expected value. In 1738, Daniel Bernoulli published an influential paper entitled Exposition of a New Theory on the Measurement of Risk, in which he uses the St. Petersburg paradox to show that expected value theory must be normatively wrong. He gives an example in which a Dutch merchant is trying to decide whether to insure a cargo being sent from Amsterdam to St Petersburg in winter.
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The fundamental reason why all martingale-type betting systems fail is that no amount of information about the results of past bets can be used to predict the results of a future bet with accuracy better than chance. In mathematical terminology, this corresponds to the assumption that the win-loss outcomes of each bet are independent and identically distributed random variables, an assumption which is valid in many realistic situations. It follows from this assumption that the expected value of a series of bets is equal to the sum, over all bets that could potentially occur in the series, of the expected value of a potential bet times the probability that the player will make that bet. In most casino games, the expected value of any individual bet is negative, so the sum of many negative numbers will also always be negative.
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The fundamental theorem of poker is a principle first articulated by David Sklansky that he believes expresses the essential nature of poker as a game of decision-making in the face of incomplete information. The fundamental theorem is stated in common language, but its formulation is based on mathematical reasoning. Each decision that is made in poker can be analyzed in terms of the expected value of the payoff of a decision. The correct decision to make in a given situation is the decision that has the largest expected value.
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In probability theory, the optional stopping theorem (or Doob's optional sampling theorem) says that, under certain conditions, the expected value of a martingale at a stopping time is equal to its initial expected value. Since martingales can be used to model the wealth of a gambler participating in a fair game, the optional stopping theorem says that, on average, nothing can be gained by stopping play based on the information obtainable so far (i.e., without looking into the future). Certain conditions are necessary for this result to hold true.
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A risk-averse contestant will choose no door and accept the guaranteed $500, while a risk-loving contestant will derive utility from the uncertainty and will therefore choose a door. If too many contestants are risk averse, the game show may encourage selection of the riskier choice (gambling on one of the doors) by offering a positive risk premium. If the game show offers $1,600 behind the good door, increasing to $800 the expected value of choosing between doors 1 and 2, the risk premium becomes $300 (i.e., $800 expected value minus $500 guaranteed amount).
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The St. Petersburg paradox or St. Petersburg lottery is a paradox related to probability and decision theory in economics. It is based on a particular (theoretical) lottery game that leads to a random variable with infinite expected value (i.e., infinite expected payoff) but nevertheless seems to be worth only a very small amount to the participants. The St. Petersburg paradox is a situation where a naive decision criterion which takes only the expected value into account predicts a course of action that presumably no actual person would be willing to take.
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The reason for its irrelevance is that maximizing the expected value of utility u(c)=(1-e^{-a c})/a gives the same result for the choice variable as does maximizing the expected value of u(c)=-e^{-a c}/a; since expected values of utility (as opposed to the utility function itself) are interpreted ordinally instead of cardinally, the range and sign of the expected utility values are of no significance. The exponential utility function is a special case of the hyperbolic absolute risk aversion utility functions.
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To call the increments stationary means that the probability distribution of any increment Xt − Xs depends only on the length t − s of the time interval; increments on equally long time intervals are identically distributed. If X is a Wiener process, the probability distribution of Xt − Xs is normal with expected value 0 and variance t − s. If X is the Poisson process, the probability distribution of Xt − Xs is a Poisson distribution with expected value λ(t − s), where λ > 0 is the "intensity" or "rate" of the process.
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Therefore, the expected value of following a certain religion could be negative. Or, one could also argue that there are an infinite number of mutually exclusive religions (which is a subset of the set of all possible religions), and that the probability of any one of them being true is zero; therefore, the expected value of following a certain religion is zero. Pascal considers this type of objection briefly in the notes compiled into the Pensées, and dismisses it as obviously wrong and disingenuous:Wetsel, David (1994). Pascal and Disbelief: Catechesis and Conversion in the Pensées.
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For general utility functions, however, expected utility analysis does not permit the expression of preferences to be separated into two parameters with one representing the expected value of the variable in question and the other representing its risk.
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Thus, the total expected value for each application of the betting system is (0.978744 − 1.339118) = −0.360374 . In a unique circumstance, this strategy can make sense. Suppose the gambler possesses exactly 63 units but desperately needs a total of 64.
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This law is not a trivial result of definitions as it might at first appear, but rather must be proved.Virtual Laboratories in Probability and Statistics, Sect. 3.1 "Expected Value: Definition and Properties", item "Basic Results: Change of Variables Theorem".
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If is the field of real numbers, then this is the probability-generating function of the probability distribution of . Similarly, () and () yield and, for every sequence , The quantity on the left-hand side of () is the expected value of .
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The expected value (E(k), E = expected) and the variance (Var(k)) are parameters used to describe the theoretical distribution of the detected photon numbers. These parameters cannot be determined directly but they can be estimated from the measurements of intensity.
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The St. Petersburg paradox (named after the journal in which Bernoulli's paper was published) arises when there is no upper bound on the potential rewards from very low probability events. Because some probability distribution functions have an infinite expected value, an expected-wealth maximizing person would pay an arbitrarily large finite amount to take this gamble. In real life, people do not do this. Bernoulli proposed a solution to this paradox in his paper: the utility function used in real life means that the expected utility of the gamble is finite, even if its expected value is infinite.
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VoC is derived strictly following its definition as the monetary amount that is big enough to just offset the additional benefit of getting more information. In other words; VoC is calculated iteratively until ::"value of decision situation with perfect information while paying VoC" = "value of current decision situation". A special case is when the decision- maker is risk neutral where VoC can be simply computed as ::VoC = "value of decision situation with perfect information" - "value of current decision situation". This special case is how expected value of perfect information and expected value of sample information are calculated where risk neutrality is implicitly assumed.
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The convexity can be used to interpret derivative pricing: mathematically, convexity is optionality – the price of an option (the value of optionality) corresponds to the convexity of the underlying payout. In Black–Scholes pricing of options, omitting interest rates and the first derivative, the Black–Scholes equation reduces to \Theta = -\Gamma, "(infinitesimally) the time value is the convexity". That is, the value of an option is due to the convexity of the ultimate payout: one has the option to buy an asset or not (in a call; for a put it is an option to sell), and the ultimate payout function (a hockey stick shape) is convex – "optionality" corresponds to convexity in the payout. Thus, if one purchases a call option, the expected value of the option is higher than simply taking the expected future value of the underlying and inputting it into the option payout function: the expected value of a convex function is higher than the function of the expected value (Jensen inequality).
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The orange consumer now has given up an orange for an apple, which next year has an expected value of 1.25 oranges. Thus both appear to have benefited from the exchange on average. Mathematically, the apparent surplus is related to Jensen's inequality.Beenstock, Michael.
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It is also possible that the expected value restrictions for the class C force the probability distribution to be zero in certain subsets of S. In that case our theorem doesn't apply, but one can work around this by shrinking the set S.
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HV 2112 is listed in the OGLE catalogue as an unresolved multiple star. The proper motions and radial velocity are consistent with other SMC objects, while the parallax is negative but acceptably close to the expected value for such a distant object.
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Assets are cash values tied to specific outcomes (e.g., Candidate X will win the election) or parameters (e.g., Next quarter's revenue). The current market prices are interpreted as predictions of the probability of the event or the expected value of the parameter.
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The expected value of the sample variance isLaw and Kelton, Simulation Modeling and Analysis, 2nd Ed. McGraw-Hill (1991), p.284, . This expression can be derived from its original source in Anderson, The Statistical Analysis of Time Series, Wiley (1971), , p.448, Equation 51.
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1 - 1/2^n which approaches 1. Huygens's result is illustrated in the next section. The eventual fate of a player at a negative expected value game cannot be better than the player at a fair game, so he will go broke as well.
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In actuarial notation the probability of this event is denoted by and can be taken from a life table. Use independence to calculate the probability of intersections. Calculate and use the probabilistic version of the Schuette–Nesbitt formula () to calculate the expected value of .
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Where environmental issues are concerned, uncertainties should always be taken into consideration. The first step to developing a standard is the evaluation of the specific risk. The expected value of the occurrence of the risk must be calculated. Then, possible damage should be classified.
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The median is zero, but the expected value does not exist, and indeed the average of n such variables have the same distribution as one such variable. It does not converge in probability toward zero (or any other value) as n goes to infinity.
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In probability theory, Bennett's inequality provides an upper bound on the probability that the sum of independent random variables deviates from its expected value by more than any specified amount. Bennett's inequality was proved by George Bennett of the University of New South Wales in 1962.
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Decision-makers are assumed to make their decisions (such as, for example, portfolio allocations) so as to maximize the expected value of the utility function. Notable special cases of HARA utility functions include the quadratic utility function, the exponential utility function, and the isoelastic utility function.
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One possible reason for this tendency of buyers to indicate lower prices is their risk aversion. By contrast, sellers may assume that the market is heterogeneous enough to include buyers with potential risk neutrality and therefore adjust their price closer to a risk neutral expected value.
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In poker, pot odds are the ratio of the current size of the pot to the cost of a contemplated call.Sklansky, 1987, Glossary Pot odds are often compared to the probability of winning a hand with a future card in order to estimate the call's expected value.
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Also, it might be considered irrationalist to gamble or buy a lottery ticket, on the basis that the expected value is negative. Irrational thought was seen in Europe as part of the reaction against Continental rationalism. For example, Johann Georg Hamann is sometimes classified as an irrationalist.
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In combinatorics, the n-th Bell number is the number of partitions of a set of size n. All of the cumulants of the sequence of Bell numbers are equal to 1. The Bell numbers are the moments of the Poisson distribution with expected value 1.
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This definition is based on the statistical expected value, integrating over infinite time. The real-world situation does not allow for such time-series, in which case a statistical estimator needs to be used in its place. A number of different estimators will be presented and discussed.
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But fair bets are, by definition, the result of comparing a gamble with an expected value of zero to some other gamble. Although it is impossible to model attitudes toward risk if one doesn't quantify utility, the theory should not be interpreted as measuring strength of preference under certainty.
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For example. if the function is defined as :g(x)= P(A\mid X=x), then :P(A\mid X) =g\circ X. Note that and are now both random variables. From the law of total probability, the expected value of is equal to the unconditional probability of .
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Tail value at risk (TVaR), also known as tail conditional expectation (TCE) or conditional tail expectation (CTE), is a risk measure associated with the more general value at risk. It quantifies the expected value of the loss given that an event outside a given probability level has occurred.
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In mathematics, a Feller-continuous process is a continuous-time stochastic process for which the expected value of suitable statistics of the process at a given time in the future depend continuously on the initial condition of the process. The concept is named after Croatian-American mathematician William Feller.
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This value of 30 represents the amount of profit for the bookmaker if he gets bets in good proportions on each of the horses. For example, if he takes £60, £50, and £20 of stakes respectively for the three horses, he receives £130 in wagers but only pays £100 back (including stakes), whichever horse wins. And the expected value of his profit is positive even if everybody bets on the same horse. The art of bookmaking is in setting the odds low enough so as to have a positive expected value of profit while keeping the odds high enough to attract customers, and at the same time attracting enough bets for each outcome to reduce his risk exposure.
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However, if the opponent has a weaker hand, betting may be the only way to get the opponent's money into the pot, as checking allows the opponent the opportunity to check in turn. It can also refer to a situation where one is faced with a bet and is considering raising instead of calling. If the betting player has a polarized range indicating that he is either betting with the nuts or is bluffing, raising with a made hand is a negative freeroll, since the expected value of calling and raising are identical when the betting player has a bluff, but the expected value of a raise is worse than a call when the bettor has the nuts.
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CHAPTER 11, WHAT TO EXPECT WHEN YOU’RE EXPECTING TO WIN THE LOTTERY: This chapter discusses the different probabilities of winning the lottery and expected value as it relates to lottery tickets, including the story of how MIT students managed to “win” the lottery every time in their town. Ellenberg also talks about the Law of Large numbers again, as well as introducing the Additivity of expected value and the games of Franc-Carreau or the “needle/noodle problem”. Many mathematicians and other famous people are mentioned in this chapter, including Georges-Louis LeClerc, Comte de Buffon, and James Harvey. CHAPTER 12, MISS MORE PLANES: The mathematical concepts in this chapter include Utility and Utils, and the Laffer curve again.
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The forecast error (also known as a residual) is the difference between the actual value and the forecast value for the corresponding period: :\ E_t = Y_t - F_t where E is the forecast error at period t, Y is the actual value at period t, and F is the forecast for period t. A good forecasting method will yield residuals that are uncorrelated. If there are correlations between residual values, then there is information left in the residuals which should be used in computing forecasts. This can be accomplished by computing the expected value of a residual as a function of the known past residuals, and adjusting the forecast by the amount by which this expected value differs from zero.
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Expected value of including uncertainty (EVIU) is a similar concept focusing on information and decision making. An EVIU can be incorporated into BDL or even represent the basis. The complementary contribution of BDL is defined by the multiplicity of the consumer of the object. The multiplicity results in a bulk dispatch.
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In a casino, the expected value is negative, due to the house's edge. The likelihood of catastrophic loss may not even be very small. The bet size rises exponentially. This, combined with the fact that strings of consecutive losses actually occur more often than common intuition suggests, can bankrupt a gambler quickly.
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Increased savings in the current period raises the expected value of future consumption. Hence the consumer reacts to increased income riskiness by raising level of saving. Yet increases in saving will also increase the variability (variance) of future consumption. This in turn gives rise to two conflicting tendencies of income and substitution effects.
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These further studies have given rise to two prominent forms of the LLN. One is called the "weak" law and the other the "strong" law, in reference to two different modes of convergence of the cumulative sample means to the expected value; in particular, as explained below, the strong form implies the weak.
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In mathematics -- specifically, in stochastic analysis -- Dynkin's formula is a theorem giving the expected value of any suitably smooth statistic of an Itō diffusion at a stopping time. It may be seen as a stochastic generalization of the (second) fundamental theorem of calculus. It is named after the Russian mathematician Eugene Dynkin.
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Data transformation may be used as a remedial measure to make data suitable for modeling with linear regression if the original data violates one or more assumptions of linear regression. For example, the simplest linear regression models assume a linear relationship between the expected value of Y (the response variable to be predicted) and each independent variable (when the other independent variables are held fixed). If linearity fails to hold, even approximately, it is sometimes possible to transform either the independent or dependent variables in the regression model to improve the linearity. For example, addition of quadratic functions of the original independent variables may lead to a linear relationship with expected value of Y, resulting in a polynomial regression model, a special case of linear regression.
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If x[n] is an infinite sequence of samples of a sample function of a wide-sense stationary process, then it is not a member of any \scriptstyle\ell^p or Lp space, with probability 1; that is, the infinite sum of samples raised to a power p does not have a finite expected value. Nevertheless, the interpolation formula converges with probability 1. Convergence can readily be shown by computing the variances of truncated terms of the summation, and showing that the variance can be made arbitrarily small by choosing a sufficient number of terms. If the process mean is nonzero, then pairs of terms need to be considered to also show that the expected value of the truncated terms converges to zero.
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In finance, a common problem is to choose a portfolio when there are two conflicting objectives — the desire to have the expected value of portfolio returns be as high as possible, and the desire to have risk, often measured by the standard deviation of portfolio returns, be as low as possible. This problem is often represented by a graph in which the efficient frontier shows the best combinations of risk and expected return that are available, and in which indifference curves show the investor's preferences for various risk-expected return combinations. The problem of optimizing a function of the expected value (first moment) and the standard deviation (square root of the second central moment) of portfolio return is called a two-moment decision model.
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So, this team is definitely unique (they contribute greatly to the Variance of the Hypothetical Mean). So we can rate this team's experience with a fairly high credibility. They often/always score a lot (low Expected Value of Process Variance) and not many teams score as much as them (high Variance of Hypothetical Mean).
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For the choice of the appropriate branch of the relation with respect to function continuity a modified version of the arctangent function is helpful. It brings in previous knowledge about the expected value by a parameter. The modified arctangent function is defined as: :. It produces a value that is as close to as possible.
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The kind of risk analysis pioneered there has become common today in fields like nuclear power, aerospace and the chemical industry. In statistical decision theory, the risk function is defined as the expected value of a given loss function as a function of the decision rule used to make decisions in the face of uncertainty.
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It is not necessary that he follow the precise rule, just that he increase his bet fast enough as he wins. This is true even if the expected value of each bet is positive. The gambler playing a fair game (with 0.5 probability of winning) will eventually either go broke or double his wealth.
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In most of the models the players are regarded to be risk neutral. This means that they intend to maximize their expected profit (or minimize their expected costs). However, some studies regard risk averse players who want to find an acceptable trade-off considering both the expected value and the variance of the profit.
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In economics, decision-making under uncertainty is often modelled using the von Neumann–Morgenstern utility function of the uncertain variable of interest, such as end-of-period wealth. Since the value of this variable is uncertain, so is the value of the utility function; it is the expected value of utility that is maximized.
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Approximately 25 per cent of the ocean surface has ample macronutrients, with little plant biomass (as defined by chlorophyll). The production in these high-nutrient low-chlorophyll (HNLC) waters is primarily limited by micronutrients especially iron. The cost of distributing iron over large ocean areas is large compared with the expected value of carbon credits.
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Violations to this assumption result in a large reduction in power. Suggested solutions to this violation are: delete a variable, combine levels of one variable (e.g., put males and females together), or collect more data. 3\. The logarithm of the expected value of the response variable is a linear combination of the explanatory variables.
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The probability of that occurring in our example is 0.0437. The second method involves computing the probability that the deviation from the expected value is as unlikely or more unlikely than the observed value, i.e. from a comparison of the probability density functions. This can create a subtle difference, but in this example yields the same probability of 0.0437.
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Consider an experiment in which a fair die is rolled 20 times. Each roll will produce one whole number between 1 and 6, and the hypothesized mean value is 3.5. The results of the rolls are then averaged together, and the mean is reported as 3.48. This is close to the expected value, and appears to support the hypothesis.
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The units used are typically hours or lifecycles. This critical relationship between a system's MTBF and its failure rate allows a simple conversion/calculation when one of the two quantities is known and an exponential distribution (constant failure rate, i.e., no systematic failures) can be assumed. The MTBF is the expected value, average or mean of the exponential distribution.
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There is a trivial randomized truthful mechanism for fair cake-cutting: select a single agent uniformly at random, and give him/her the entire cake. This mechanism is trivially truthful because it asks no questions. Moreover, it is fair in expectation: the expected value of each partner is exactly 1/n. However, the resulting allocation is not fair.
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Although scoring rules are introduced in probabilistic forecasting literature, the definition is general enough to consider non-probabilistic measures such as mean absolute error or mean square error as some specific scoring rules. The main characteristic of such scoring rules is S(G,y) is just a function of the expected value of G (i.e., E(G)).
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This can be expressed more precisely by the notion of expected value, which is uniformly negative (from the player's perspective). This advantage is called the house edge. In games such as poker where players play against each other, the house takes a commission called the rake. Casinos sometimes give out complimentary items or comps to gamblers.
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RAIM detects faults with redundant GPS pseudorange measurements. That is, when more satellites are available than needed to produce a position fix, the extra pseudoranges should all be consistent with the computed position. A pseudorange that differs significantly from the expected value (i.e., an outlier) may indicate a fault of the associated satellite or another signal integrity problem (e.g.
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This is a generalization of the notion of a probability measure, where the probability axiom of countable additivity is weakened. A capacity is used as a subjective measure of the likelihood of an event, and the "expected value" of an outcome given a certain capacity can be found by taking the Choquet integral over the capacity.
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Cells that project from the lateral amygdala to the central amygdala allow for the initiation of an emotional response if a stimulus is deemed threatening. Cognitive Control. Evaluating a gamble and calculating its expected value requires a certain amount of cognitive control. Several brain areas are dedicated to monitoring the congruence between expected and actual outcomes.
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A continuity correction can also be applied when other discrete distributions supported on the integers are approximated by the normal distribution. For example, if X has a Poisson distribution with expected value λ then the variance of X is also λ, and :P(X\leq x)=P(X if Y is normally distributed with expectation and variance both λ.
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The random walk hypothesis considers that asset prices in an organized market evolve at random, in the sense that the expected value of their change is zero but the actual value may turn out to be positive or negative. More generally, asset prices are influenced by a variety of unpredictable events in the general economic environment.
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Convergence of the Metropolis–Hastings algorithm. Markov chain Monte Carlo attempts to approximate the blue distribution with the orange distribution. Markov chain Monte Carlo methods create samples from a continuous random variable, with probability density proportional to a known function. These samples can be used to evaluate an integral over that variable, as its expected value or variance.
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A picking sequence is a simple protocol where the agents take turns in selecting items, based on some pre-specified sequence of turns. The goal is to design the picking-sequence in a way that maximizes the expected value of a social welfare function (e.g. egalitarian or utilitarian) under some probabilistic assumptions on the agents' valuations.
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Comparison Of Various Pulmonary Function Parameters In The Diagnosis Of Obstructive Lung Disease In Patients With Normal Fev1/FVC And Low FVC. American Journal of Respiratory and Critical Care Medicine, 191, American Journal of Respiratory and Critical Care Medicine, 2015, Vol.191. One definition requires a total lung capacity which is 80% or less of the expected value.
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In estimation theory and decision theory, a Bayes estimator or a Bayes action is an estimator or decision rule that minimizes the posterior expected value of a loss function (i.e., the posterior expected loss). Equivalently, it maximizes the posterior expectation of a utility function. An alternative way of formulating an estimator within Bayesian statistics is maximum a posteriori estimation.
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In portfolio theory, the locus of mean-variance efficient portfolios (called the efficient frontier) is the upper half of the east-opening branch of a hyperbola drawn with the portfolio return's standard deviation plotted horizontally and its expected value plotted vertically; according to this theory, all rational investors would choose a portfolio characterized by some point on this locus.
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Daniel Bernoulli proposed that a nonlinear function of utility of an outcome should be used instead of the expected value of an outcome, accounting for risk aversion, where the risk premium is higher for low-probability events than the difference between the payout level of a particular outcome and its expected value. Bernoulli further proposed that it was not the goal of the gambler to maximize his expected gain but to instead maximize the logarithm of his gain. Bernoulli's paper was the first formalization of marginal utility, which has broad application in economics in addition to expected utility theory. He used this concept to formalize the idea that the same amount of additional money was less useful to an already- wealthy person than it would be to a poor person.
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Research suggests that people do not evaluate prospects by the expected value of their monetary outcomes, but rather by the expected value of the subjective value of these outcomes (see also Expected utility). In most real-life situations, the probabilities associated with each outcome are not specified by the situation, but have to be subjectively estimated by the decision-maker. The subjective value of a gamble is again a weighted average, but now it is the subjective value of each outcome that is weighted by its probability. To explain risk aversion within this framework, Bernoulli proposed that subjective value, or utility, is a concave function of money. In such a function, the difference between the utilities of $200 and $100, for example, is greater than the utility difference between $1,200 and $1,100.
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Robust measures of scale can be used as estimators of properties of the population, either for parameter estimation or as estimators of their own expected value. For example, robust estimators of scale are used to estimate the population variance or population standard deviation, generally by multiplying by a scale factor to make it an unbiased consistent estimator; see scale parameter: estimation. For example, dividing the IQR by 2 erf−1(1/2) (approximately 1.349), makes it an unbiased, consistent estimator for the population standard deviation if the data follow a normal distribution. In other situations, it makes more sense to think of a robust measure of scale as an estimator of its own expected value, interpreted as an alternative to the population variance or standard deviation as a measure of scale.
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The expected utility hypothesis is that rationality can be modeled as maximizing an expected value, which given the theorem, can be summarized as "rationality is VNM-rationality". However, the axioms themselves have been critiqued on various grounds, resulting in the axioms being given further justification.Peterson, Chapter 8. VNM-utility is a decision utility in that it is used to describe decision preferences.
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The Cauchy distribution is often used in statistics as the canonical example of a "pathological" distribution since both its expected value and its variance are undefined (but see below). The Cauchy distribution does not have finite moments of order greater than or equal to one; only fractional absolute moments exist., Chapter 16. The Cauchy distribution has no moment generating function.
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Let one round be defined as a sequence of consecutive losses followed by either a win, or bankruptcy of the gambler. After a win, the gambler "resets" and is considered to have started a new round. A continuous sequence of martingale bets can thus be partitioned into a sequence of independent rounds. Following is an analysis of the expected value of one round.
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For some probability distributions, the expected value may be infinite or undefined, but if defined, it is unique. The mean of a (finite) sample is always defined. The median is the value such that the fractions not exceeding it and not falling below it are each at least 1/2. It is not necessarily unique, but never infinite or totally undefined.
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Out of collaborations in Chance News with Charles M. Grinstead and William P. Peterson, a book Probability Tales (2011) was published by American Mathematical Society in the Student Mathematical Library. The book covers four topics: streaks in sports as streaks of successful Bernoulli trials (like hitting streaks), constructing stock market models, estimating expected value of a lottery ticket, and reliability of fingerprint identification.
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There have been many variations of this problem including the case of a uniformly random set of points, for which arguments based on either Kolmogorov complexity or Poisson approximation show that the expected value of the minimum area is inversely proportional to the cube of the number of points... Variations involving the volume of higher-dimensional simplices have also been studied..
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This is the probability mass function of the Poisson distribution with expected value λ. Note that if the probability density function is a function of various parameters, so too will be its normalizing constant. The parametrised normalizing constant for the Boltzmann distribution plays a central role in statistical mechanics. In that context, the normalizing constant is called the partition function.
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The most general of these are the accessibility, which uses the diversity of random walks to measure how accessible the rest of the network is from a given start node, and the expected force, derived from the expected value of the force of infection generated by a node. Both of these measures can be meaningfully computed from the structure of the network alone.
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Ming Li and Paul Vitanyi, An Introduction to Kolmogorov Complexity and Its Applications. Springer-Verlag, N.Y., 2008p 339 ff. All computable theories which perfectly describe previous observations are used to calculate the probability of the next observation, with more weight put on the shorter computable theories. Marcus Hutter's universal artificial intelligence builds upon this to calculate the expected value of an action.
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Suppose we wish to multiply two numbers each with n bits of precision. Using the typical long multiplication method, we need to perform n^2 operations. With stochastic computing, we can AND together any number of bits and the expected value will always be correct. (However, with a small number of samples the variance will render the actual result highly inaccurate).
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The q-value can be interpreted as the false discovery rate (FDR): the proportion of false positives among all positive results. Given a set of test statistics and their associated q-values, rejecting the null hypothesis for all tests whose q-value is less than or equal to some threshold \alpha ensures that the expected value of the false discovery rate is \alpha.
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The Catholic Church sees merit in examining what it calls "partial agnosticism", specifically those systems that "do not aim at constructing a complete philosophy of the unknowable, but at excluding special kinds of truth, notably religious, from the domain of knowledge". However, the Church is historically opposed to a full denial of the capacity of human reason to know God. The Council of the Vatican declares, "God, the beginning and end of all, can, by the natural light of human reason, be known with certainty from the works of creation". Blaise Pascal argued that even if there were truly no evidence for God, agnostics should consider what is now known as Pascal's Wager: the infinite expected value of acknowledging God is always greater than the finite expected value of not acknowledging his existence, and thus it is a safer "bet" to choose God.
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They succeeded in detecting the enzyme activity from the microsomal fraction. This was the crucial step in the serendipitous discovery of lysosomes. To estimate this enzyme activity, they used that of the standardized enzyme acid phosphatase and found that the activity was only 10% of the expected value. One day, the enzyme activity of purified cell fractions which had been refrigerated for five days was measured.
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The measured half-life is close to the expected value for ground state isomer, 277Hs. Further research is required to confirm the production of the isomer. A more recent study suggests that this observed activity may actually be from 278Bh. ;269Hs The direct synthesis of 269Hs has resulted in the observation of three alpha particles with energies 9.21, 9.10, and 8.94 MeV emitted from 269Hs atoms.
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Integrals of this type appear frequently when calculating electronic properties, like the heat capacity, in the free electron model of solids. In these calculations the above integral expresses the expected value of the quantity H(\varepsilon). For these integrals we can then identify \beta as the inverse temperature and \mu as the chemical potential. Therefore, the Sommerfeld expansion is valid for large \beta (low temperature) systems.
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A certain loss is viewed more negatively than an uncertain loss with the same expected value. Closely related to this is people tendency to be loss averse. They view saving a statistical life as a gain, whereas saving an identifiable victim is seen as avoiding a loss. Together, these effects result in people being more likely to aid identifiable, certain victims than statistical, uncertain victims.
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A method has been developed to address the bias due to the phenomenon. Stander and Stander develop a method which involves introducing a transition matrix between the two sets and taking a probability weighted expected value Stander, M and Stander, J. "A simple method for correcting for the Will Rogers phenomenon with biometrical applications", Biometrical Journal, 20 January 2020. Retrieved on 02 July 2020..
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Nicolas Bernoulli himself proposed an alternative idea for solving the paradox. He conjectured that people will neglect unlikely events . Since in the St. Petersburg lottery only unlikely events yield the high prizes that lead to an infinite expected value, this could resolve the paradox. The idea of probability weighting resurfaced much later in the work on prospect theory by Daniel Kahneman and Amos Tversky.
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Various authors, including Jean le Rond d'Alembert and John Maynard Keynes, have rejected maximization of expectation (even of utility) as a proper rule of conduct. Keynes, in particular, insisted that the relative risk of an alternative could be sufficiently high to reject it even if its expectation were enormous. Recently, some researchers have suggested to replace the expected value by the median as the fair value.
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Recently some authors suggested using heuristic parameters (e.g. assessing the possible gains without neglecting the risks of the Saint Petersburg lottery) because of the highly stochastic context of this game . The expected output should therefore be assessed in the limited period where we can likely make our choices and, besides the non-ergodic features , considering some inappropriate consequences we could attribute to the expected value .
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Myron Tribus (1961) Thermodynamics and Thermostatics: An Introduction to Energy, Information and States of Matter, with Engineering Applications (D. Van Nostrand, 24 West 40 Street, New York 18, New York, U.S.A) Tribus, Myron (1961), pp. 64-66 borrow. When the event is a random realization (of a variable) the self-information of the variable is defined as the expected value of the self-information of the realization.
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In the natural sciences, especially in atmospheric and Earth sciences involving applied statistics, an anomaly is a persisting deviation in a quantity from its expected value, e.g., the systematic difference between a measurement and a trend or a model prediction.Wilks, D.S. (1995) Statistical Methods in the Atmospheric science, Academic Press. (page 42) Similarly, a standardized anomaly equals an anomaly divided by a standard deviation.
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In both cases the intersection should be a point, because, again, if one cycle is moved, this would be the intersection. The intersection of two cycles and is called proper if the codimension of the (set-theoretic) intersection is the sum of the codimensions of and , respectively, i.e. the "expected" value. Therefore, the concept of moving cycles using appropriate equivalence relations on algebraic cycles is used.
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Going back to the original rental-harmony problem, it is possible to consider randomized mechanisms. A randomized mechanism returns a probability distribution over room-assignments and rent-divisions. A randomized mechanism is truthful in expectation if no partner can increase the expected value of his utility by mis-reporting his valuations to the rooms. The fairness of a randomized mechanism can be measured in several ways: 1\.
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In statistics, the Lilliefors test is a normality test based on the Kolmogorov–Smirnov test. It is used to test the null hypothesis that data come from a normally distributed population, when the null hypothesis does not specify which normal distribution; i.e., it does not specify the expected value and variance of the distribution. It is named after Hubert Lilliefors, professor of statistics at George Washington University.
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VoC is often illustrated using the example of paying for a consultant in a business transaction, who may either be perfect (expected value of perfect information) or imperfect (expected value of imperfect information). In a typical consultant situation, the consultant would be paid up to cost c for their information, based on the expected cost E without the consultant and the revised cost F with the consultant's information. In a perfect information scenario, E can be defined as the sum product of the probability of a good outcome g times its cost k, plus the probability of a bad outcome (1-g) times its cost k'>k: E = gk + (1-g)k', which is revised to reflect expected cost F of perfect information including consulting cost c. The perfect information case assumes the bad outcome does not occur due to the perfect information consultant.
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When a player holds a drawing hand (a hand that is behind now but is likely to win if a certain card is drawn) pot odds are used to determine the expected value of that hand when the player is faced with a bet. The expected value of a call is determined by comparing the pot odds to the odds of drawing a card that wins the pot. When the odds of drawing a card that wins the pot are numerically higher than the pot odds, the call has a positive expectation; on average, a portion of the pot that is greater than the cost of the call is won. Conversely, if the odds of drawing a winning card are numerically lower than the pot odds, the call has a negative expectation, and the expectation is to win less money on average than it costs to call the bet.
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The disadvantages of the GnRH stimulation test is it takes a long time to perform and requires multiple collections from the patient, making the process time consuming and inconvenient. The test is highly specific but has low sensitivity as the LH hormone response is usually observed in later stages of CPP. There are also overlaps in the expected value in the GnRH test results of individuals with CPP and PT.
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Another application of the law of averages is a belief that a sample's behaviour must line up with the expected value based on population statistics. For example, suppose a fair coin is flipped 100 times. Using the law of averages, one might predict that there will be 50 heads and 50 tails. While this is the single most likely outcome, there is only an 8% chance of it occurring.
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There are 10 cells. If the null hypothesis had specified a single distribution, rather than requiring λ to be estimated, then the null distribution of the test statistic would be a chi-square distribution with 10 − 1 = 9 degrees of freedom. Since λ had to be estimated, one additional degree of freedom is lost. The expected value of a chi-square random variable with 8 degrees of freedom is 8.
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A known current is passed down the connection and the voltage that develops is measured. From the voltage and current the resistance of the connection can be calculated and compared to the expected value. There are two common ways to test for a short: # A low voltage test. A low power, low voltage source is connected between two conductors that should not be connected and the amount of current is measured.
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The plaintiff sought to acquire half of the expected value of her husband's medical degree during divorce proceedings. The plaintiff provided testimony about the earnings potential associated with a medical degree and sought half of the expected earnings associated with the degree. The court ruled that the medical degree was not a property interest subject to division, but rather simply an expectancy that may not even vest.Casner, pp.
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The kepstrum, which stands for "Kolmogorov-equation power-series time response", is similar to the cepstrum and has the same relation to it as expected value has to statistical average, i.e. cepstrum is the empirically measured quantity, while kepstrum is the theoretical quantity. It was in use before the cepstrum. "Predictive decomposition of time series with applications to seismic exploration", E. A. Robinson MIT report 1954; Geophysics 1967 vol.
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Existing futures contracts can be priced using elements of the spot-futures parity equation, where K is the settlement price of the existing contract, S_0 is the current spot price and P_0 is the (expected) value of the existing contract today: : P_0 = S_0 - K e^{-rT} which upon application of the spot-futures parity equation becomes: : P_0 = (F_0 - K)e^{-rT} Where F_0 is the forward price today.
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By the law of large numbers, integrals described by the expected value of some random variable can be approximated by taking the empirical mean (a.k.a. the sample mean) of independent samples of the variable. When the probability distribution of the variable is parametrized, mathematicians often use a Markov chain Monte Carlo (MCMC) sampler. The central idea is to design a judicious Markov chain model with a prescribed stationary probability distribution.
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In the atmospheric sciences, the climatological annual cycle is often used as the expected value. Famous atmospheric anomalies are for instance the Southern Oscillation index (SOI) and the North Atlantic oscillation index. SOI is the atmospheric component of El Niño, while NAO plays an important role for European weather by modification of the exit of the Atlantic storm track. A climate normal can also be used to deriva a climate anomaly.
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The solution is to expand the function z in a second-order Taylor series; the expansion is done around the mean values of the several variables x. (Usually the expansion is done to first order; the second-order terms are needed to find the bias in the mean. Those second-order terms are usually dropped when finding the variance; see below). 5\. With the expansion in hand, find the expected value.
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Life insurance actuaries determine the probability of death in any given year, and based on this probability determine the expected value of the loss payment. These expected future payment are discounted back to the start of the coverage period and summed to determine the net single premium. The net single premium may be leveled to convert to installment premiums. A loading for expenses is added to determine the gross premium.
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This trade-off is sometimes represented in what is called an Euler equation. A time-series path in the recursive model is the result of a series of these two-period decisions. In the neoclassical model, the consumer or producer maximizes utility (or profits). In the recursive model, the subject maximizes value or welfare, which is the sum of current rewards or benefits and discounted future expected value.
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Statistical bias is a systematic tendency in the process of data collection, which results in lopsided, misleading results. This can occur in any of a number of ways, in the way the sample is selected, or in the way data are collected. It is a property of a statistical technique or of its results whereby the expected value of the results differs from the true underlying quantitative parameter being estimated.
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Formally, the convexity adjustment arises from the Jensen inequality in probability theory: the expected value of a convex function is greater than or equal to the function of the expected value: :E[f(X)] \geq f(E[X]). Geometrically, if the model price curves up on both sides of the present value (the payoff function is convex up, and is above a tangent line at that point), then if the price of the underlying changes, the price of the output is greater than is modeled using only the first derivative. Conversely, if the model price curves down (the convexity is negative, the payoff function is below the tangent line), the price of the output is lower than is modeled using only the first derivative. The precise convexity adjustment depends on the model of future price movements of the underlying (the probability distribution) and on the model of the price, though it is linear in the convexity (second derivative of the price function).
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In practical terms, one begins with an initial guess as to the expected value of a quantity, and then, using various methods and instruments, reduces the uncertainty in the value. Note that in this view, unlike the positivist representational theory, all measurements are uncertain, so instead of assigning one value, a range of values is assigned to a measurement. This also implies that there is not a clear or neat distinction between estimation and measurement.
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When practitioners need to consider multiple models, they can specify a probability-measure on the models and then select any design maximizing the expected value of such an experiment. Such probability-based optimal-designs are called optimal Bayesian designs. Such Bayesian designs are used especially for generalized linear models (where the response follows an exponential-family distribution).Bayesian designs are discussed in Chapter 18 of the textbook by Atkinson, Donev, and Tobias.
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The exact interpretation of the Pearson measure of kurtosis (or excess kurtosis) used to be disputed, but is now settled. As Westfall notes in 2014, "...its only unambiguous interpretation is in terms of tail extremity; i.e., either existing outliers (for the sample kurtosis) or propensity to produce outliers (for the kurtosis of a probability distribution)." The logic is simple: Kurtosis is the average (or expected value) of the standardized data raised to the fourth power.
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Sufficiency finds a useful application in the Rao–Blackwell theorem, which states that if g(X) is any kind of estimator of θ, then typically the conditional expectation of g(X) given sufficient statistic T(X) is a better estimator of θ, and is never worse. Sometimes one can very easily construct a very crude estimator g(X), and then evaluate that conditional expected value to get an estimator that is in various senses optimal.
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Hubbard also argues that defining risk as the product of impact and probability presumes, unrealistically, that decision-makers are risk-neutral. A risk- neutral person's utility is proportional to the expected value of the payoff. For example, a risk-neutral person would consider 20% chance of winning $1 million exactly as desirable as getting a certain $200,000. However, most decision-makers are not actually risk-neutral and would not consider these equivalent choices.
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Risk aversion (red) contrasted to risk neutrality (yellow) and risk loving (orange) in different settings. Left graph: A risk averse utility function is concave (from below), while a risk loving utility function is convex. Middle graph: In standard deviation-expected value space, risk averse indifference curves are upward sloped. Right graph: With fixed probabilities of two alternative states 1 and 2, risk averse indifference curves over pairs of state-contingent outcomes are convex.
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Limitations to centrality measures have led to the development of more general measures. Two examples are the accessibility, which uses the diversity of random walks to measure how accessible the rest of the network is from a given start node, and the expected force, derived from the expected value of the force of infection generated by a node. Both of these measures can be meaningfully computed from the structure of the network alone.
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To promote robustness some of the system parameters may be assumed stochastic instead of deterministic. The associated more difficult control problem leads to a similar optimal controller of which only the controller parameters are different. It is possible to compute the expected value of the cost function for the optimal gains, as well as any other set of stable gains. Finally, the LQG controller is also used to control perturbed non-linear systems.
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The second equation follows since θ is measurable with respect to the conditional distribution P(x\mid\theta). An estimator is said to be unbiased if its bias is equal to zero for all values of parameter θ, or equivalently, if the expected value of the estimator matches that of the parameter. In a simulation experiment concerning the properties of an estimator, the bias of the estimator may be assessed using the mean signed difference.
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Since the expected value of an unbiased estimator is equal to the parameter value, E[T]=\theta. Therefore, MSE(T)=Var(T) as the (E[T]-\theta)^2 term drops out from being equal to 0. If an unbiased estimator of a parameter θ attains e(T) = 1 for all values of the parameter, then the estimator is called efficient. Equivalently, the estimator achieves equality in the Cramér–Rao inequality for all θ.
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Cost of Delay is "a way of communicating the impact of time on the outcomes we hope to achieve". More formally, it is the partial derivative of the total expected value with respect to time. Cost of Delay combines an understanding of value with how that value leaks away over time. It is a tactic that helps communicate and prioritize development decisions by calculating the impact of time on value creation & capture.
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Peak flow meter (made in USA) Peak flow readings are higher when patients are well, and lower when the airways are constricted. From changes in recorded values, patients and doctors may determine lung functionality, the severity of asthma symptoms, and treatment. Measurement of PEFR requires training to correctly use a meter and the normal expected value depends on the patient's sex, age, and height. It is classically reduced in obstructive lung disorders such as asthma.
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If we know the exact value of the treatment effect, there is no need to do the experiment. To address this issue, we can consider conditional power in a Bayesian setting by considering the treatment effect parameter to be a random variable. Taking the expected value of the conditional power with respect to the posterior distribution of the parameter gives the predictive power. Predictive power can also be calculated in a frequentist setting.
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If X1, ..., Xn are independent identically distributed random variables that are normally distributed, the probability distribution of their studentized range is what is usually called the studentized range distribution. Note that the definition of q does not depend on the expected value or the standard deviation of the distribution from which the sample is drawn, and therefore its probability distribution is the same regardless of those parameters. tables of the distribution quantiles are available here.
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A standard derivation for solving the Black–Scholes PDE is given in the article Black–Scholes equation. The Feynman–Kac formula says that the solution to this type of PDE, when discounted appropriately, is actually a martingale. Thus the option price is the expected value of the discounted payoff of the option. Computing the option price via this expectation is the risk neutrality approach and can be done without knowledge of PDEs.
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The first consumer-level CPU deliveries using a 22 nm process started in April 2012 with the Intel Ivy Bridge processors. The ITRS 2006 Front End Process Update indicates that equivalent physical oxide thickness will not scale below 0.5 nm (about twice the diameter of a silicon atom), which is the expected value at the 22 nm node. This is an indication that CMOS scaling in this area has reached a wall at this point, possibly disturbing Moore's law.
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The \beta coefficient should not be interpreted as the effect of x_i on y_i, as one would with a linear regression model; this is a common error. Instead, it should be interpreted as the combination of (1) the change in y_i of those above the limit, weighted by the probability of being above the limit; and (2) the change in the probability of being above the limit, weighted by the expected value of y_i if above.
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Dealing with parameter uncertainty and estimation error is a large topic in portfolio theory. The second-order Taylor polynomial can be used as a good approximation of the main criterion. Primarily, it is useful for stock investment, where the fraction devoted to investment is based on simple characteristics that can be easily estimated from existing historical data – expected value and variance. This approximation leads to results that are robust and offer similar results as the original criterion.
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In probability theory and statistics, Campbell's theorem or the Campbell–Hardy theorem is either a particular equation or set of results relating to the expectation of a function summed over a point process to an integral involving the mean measure of the point process, which allows for the calculation of expected value and variance of the random sum. One version of the theorem,D. Stoyan, W. S. Kendall, J. Mecke. Stochastic geometry and its applications, volume 2.
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Sometimes, the equivalent problem of minimizing the expected value of loss is considered, where loss is (–1) times utility. Another equivalent problem is minimizing expected regret. "Utility" is only an arbitrary term for quantifying the desirability of a particular decision outcome and not necessarily related to "usefulness." For example, it may well be the optimal decision for someone to buy a sports car rather than a station wagon, if the outcome in terms of another criterion (e.g.
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Then a fair coin is tossed to decide whether Envelope B should contain half or twice that amount, and only then given to Baba. Broome in 1995 called the probability distribution 'paradoxical' if for any given first-envelope amount x, the expectation of the other envelope conditional on x is greater than x. The literature contains dozens of commentaries on the problem, much of which observes that a distribution of finite values can have an infinite expected value.
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Trimmed estimators can also be used as statistics in their own right – for example, the median is a measure of location, and the IQR is a measure of dispersion. In these cases, the sample statistics can act as estimators of their own expected value. For example, the MAD of a sample from a standard Cauchy distribution is an estimator of the population MAD, which in this case is 1, whereas the population variance does not exist.
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The Pay-per-Share (PPS) approach offers an instant, guaranteed payout to a miner for his contribution to the probability that the pool finds a block. Miners are paid out from the pool's existing balance and can withdraw their payout immediately. This model allows for the least possible variance in payment for miners while also transferring much of the risk to the pool's operator. Each share costs exactly the expected value of each hash attempt R = B \cdot p.
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In control theory, the linear–quadratic–Gaussian (LQG) control problem is one of the most fundamental optimal control problems. It concerns linear systems driven by additive white Gaussian noise. The problem is to determine an output feedback law that is optimal in the sense of minimizing the expected value of a quadratic cost criterion. Output measurements are assumed to be corrupted by Gaussian noise and the initial state, likewise, is assumed to be a Gaussian random vector.
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Myerson and Satterthwaite study the following requirements that an ideal mechanism should satisfy (see also Double auction#requirements): 1\. ex-ante individual rationality (IR): The expected value of both Bob and Sally should be non-negative (so that they have an initial incentive to participate). Formally: U_S(v_S,v_S)\geq 0 and U_B(v_B,v_B)\geq 0. 2\. Weak balanced budget (WBB): The auctioneer should not have to bring money from home in order to subsidize the trade. 3\.
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This means that players who are having exceptionally good or exceptionally bad outcomes are more likely to gamble and continue playing than average players. The lucky or unlucky players were willing to reject offers of over one hundred percent of the expected value of their case in order to continue playing. This shows a shift from risk avoiding behavior to risk seeking behavior. This study highlights behavioral biases that are not accounted for by traditional game theory.
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That is, the willingness to pay to avoid the adverse change equates the post-change utility, diminished by the presence of the adverse change (on the right side), with utility without the adverse change but with payment having been made to avoid it. The concept extends readily to a context of uncertain outcomes, in which case the utility function above is replaced by the expected value of a von Neumann-Morgenstern utility function (See expected utility hypothesis).
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B. Fischhoff, L. D. Phillips, and S. Lichtenstein, "Calibration of Probabilities: The State of the Art to 1980," in Judgement under Uncertainty: Heuristics and Biases, ed. D. Kahneman and A. Tversky (Cambridge University Press, 1982) That is, their probabilities are neither overconfident (too high) nor underconfident (too low). # Computing the value of additional information. AIE uses information value calculations from decision theory such as the expected value of perfect information and the value of imperfect (partial) information.
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Most importantly, : ΦG(x) = Φ(x) for x ≥ . Note that Φ() = 0.958…, thus the classical 95% confidence interval for the unknown expected value of Gaussian distributions covers the center of symmetry with at least 95% probability for Gaussian scale mixture distributions. On the other hand, the 90% quantile of ΦG(x) is 4/5 = 1.385… > Φ−1(0.9) = 1.282… The following critical values are important in applications: 0.95 = Φ(1.645) = ΦG(1.651), and 0.9 = Φ(1.282) = ΦG(1.386).
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Operators are also involved in probability theory, such as expectation, variance, and covariance. Indeed, every covariance is basically a dot product; every variance is a dot product of a vector with itself, and thus is a quadratic norm; every standard deviation is a norm (square root of the quadratic norm); the corresponding cosine to this dot product is the Pearson correlation coefficient; expected value is basically an integral operator (used to measure weighted shapes in the space).
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In the biased estimator, by using the sample mean instead of the true mean, you are underestimating each xi − µ by x − µ. We know that the variance of a sum is the sum of the variances (for uncorrelated variables). So, to find the discrepancy between the biased estimator and the true variance, we just need to find the expected value of (x − µ)2. This is just the variance of the sample mean, which is σ2/n.
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This level of testing usually requires thorough test cases to be provided to the tester, who then can simply verify that for a given input, the output value (or behavior), either "is" or "is not" the same as the expected value specified in the test case. Test cases are built around specifications and requirements, i.e., what the application is supposed to do. It uses external descriptions of the software, including specifications, requirements, and designs to derive test cases.
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Deviations from this expected value are indicative of other processes that affect the delta ratio of radiocarbon, namely radioactive decay. This deviation can be converted to a time, giving the age of the water at that location. Doing this over the world's ocean can yield a circulation pattern of the ocean and the rate at which water flow through the deep ocean. Using this circulation in conjunction with the surface circulation allows scientists to understand the energy balance of the world.
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The RMS over all time of a periodic function is equal to the RMS of one period of the function. The RMS value of a continuous function or signal can be approximated by taking the RMS of a sample consisting of equally spaced observations. Additionally, the RMS value of various waveforms can also be determined without calculus, as shown by Cartwright. In the case of the RMS statistic of a random process, the expected value is used instead of the mean.
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The phenomenon of calor licitantis is believed to be as old as auctions themselves. This term was first used in the court system of Rome to describe the irrational behavior of bidders at auctions. The use of the phrase seemed to describe both the mental state of the bidder and the result of that state; specifically, that through the bidding process undertaken by one suffering from calor licitantis, the price of an item was driven above and beyond its typical or expected value.
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MALT consists of two programs: malt-build and malt-run. Malt-build is used to construct an index for the given database of reference sequences. Instead, malt-run is used to align a set of query sequences against the reference database. The program then computes the bit-score and the expected value (E-value) of the alignment and decides whether to keep or discard the alignment depending on user- specified thresholds for the bit-score, the E-value or the per cent identity.
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In the context of the theory of the firm, a risk neutral firm facing risk about the market price of its product, and caring only about profit, would maximize the expected value of its profit (with respect to its choices of labor input usage, output produced, etc.). But a risk averse firm in the same environment would typically take a more cautious approach.Sandmo, Agnar. "On the theory of the competitive firm under price uncertainty," American Economic Review 61, March 1971, 65-73.
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Legal financing is a fairly recent phenomenon in the United States, beginning in or around 1997. Litigation funding is available in most U.S. jurisdictions. Litigation funding is most commonly sought in personal injury cases, but may also be sought for commercial disputes, civil rights cases, and workers' compensation cases. The amount of money that plaintiffs receive through legal financing varies widely, but often is around 10 to 15 percent of the expected value of judgment or settlement of their lawsuit.
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NOPs are often involved when cracking software that checks for serial numbers, specific hardware or software requirements, presence or absence of hardware dongles, etc. This process is accomplished by altering functions and subroutines to bypass security checks and instead simply return the expected value being checked for. Because most of the instructions in the security check routine will be unused, these would be replaced with NOPs, thus removing the software's security functionality without altering the positioning of everything which follows in the binary.
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The quadratic loss function is also used in linear-quadratic optimal control problems. In these problems, even in the absence of uncertainty, it may not be possible to achieve the desired values of all target variables. Often loss is expressed as a quadratic form in the deviations of the variables of interest from their desired values; this approach is tractable because it results in linear first-order conditions. In the context of stochastic control, the expected value of the quadratic form is used.
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The costs of over- or undercontracting and then selling or buying power in the balancing market are typically so high that they can lead to huge financial losses and bankruptcy in the extreme case. In this respect electric utilities are the most vulnerable, since they generally cannot pass their costs on to the retail customers. While there have been a variety of empirical studies on point forecasts (i.e., the "best guess" or expected value of the spot price), probabilistic - i.e.
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Fixer Upper became popular soon after its debut, and the series is largely credited with the rise in popularity of "Farmhouse-chic" interior design in the late 2010s. In 2018, Zillow reported that homes with architectural features mentioned on the show, such as wainscoting, shiplap, clawfoot bathtubs, and barn doors, sold at an average of 30 percent above expected value. In addition, the show has generated an increase in tourism and economic development in Waco, Texas, where the show was filmed.
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In mathematics, the Bhatia–Davis inequality, named after Rajendra Bhatia and Chandler Davis, is an upper bound on the variance σ2 of any bounded probability distribution on the real line. Suppose a distribution has minimum m, maximum M, and expected value μ. Then the inequality says: : \sigma^2 \le (M - \mu)(\mu - m). \, Equality holds precisely if all of the probability is concentrated at the endpoints m and M. The Bhatia–Davis inequality is stronger than Popoviciu's inequality on variances.
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Suppose a game show participant may choose one of two doors, one that hides $1,000 and one that hides $0. Further suppose that the host also allows the contestant to take $500 instead of choosing a door. The two options (choosing between door 1 and door 2, or taking $500) have the same expected value of $500, so no risk premium is being offered for choosing the doors rather than the guaranteed $500. A contestant unconcerned about risk is indifferent between these choices.
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These models include the arbitrage pricing theory and the modern portfolio theory families of models. The other models that incorporate macro risk data are valuation models or the closely related fundamental analysis models. Used primarily by those focusing on longer term investments including wealth managers, financial planners, and some institutional investors, these models are examples of intrinsic value analysis. In such analysis, forecasts of future company earnings are used to estimate the current and expected value of the investment being studied.
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A player may be hampered by bad luck in backgammon, Monopoly, or Risk; but over many games a skilled player will win more often. The elements of luck can also make for more excitement at times, and allow for more diverse and multifaceted strategies, as concepts such as expected value and risk management must be considered. Luck may be introduced into a game by a number of methods. The use of dice of various sorts goes back to the earliest board games.
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Dopamine neurons are thought to be involved in learning to predict which behaviours will lead to a reward (for example food or sex). In particular, it is suggested that dopamine neurons fire when a reward is greater than that previously expected; a key component of many reinforcement learning models. This signal can then be used to update the expected value of that action. Many recreational drugs, such as cocaine, mimic this reward response—providing an explanation for their addictive nature.
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In fact, even if all estimates have astronomical absolute values for their errors, if the expected value of the error is zero, the estimator is unbiased. Also, an estimator's being biased does not preclude the error of an estimate from being zero in a particular instance. The ideal situation is to have an unbiased estimator with low variance, and also try to limit the number of samples where the error is extreme (that is, have few outliers). Yet unbiasedness is not essential.
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In the latter case no price level drift is allowed away from the predetermined path, while in the former case any stochastic change to the price level permanently affects the expected values of the price level at each time along its future path. In either case the price level has drift in the sense of a rising expected value, but the cases differ according to the type of non- stationarity: difference stationarity in the former case, but trend stationarity in the latter case.
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To estimate the exact enzyme activity, the team adopted a procedure using a standardised enzyme acid phosphatase; but they were finding the activity was unexpectedly low—quite low, i.e., some 10% of the expected value. Then one day they measured the enzyme activity of some purified cell fractions that had been stored for five days. To their surprise the enzyme activity was increased back to that of the fresh sample; and similar results were replicated every time the procedure was repeated.
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Seal of Manaus Another cache of latex was collected and then a tribal party carried it overland to their destination: the trading post on the banks of the Río Purús. Halting nearby, they built a raft to float the latex the rest of the way; as usual, Córdova put on clothes and took the rubber in alone. Its price had continued its decline, and the latex had little of its expected value. Docked by the outpost was the Filó, a Brazilian launch.
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An overview of gaming appeared in Gambling for Dummies (co-authored with Richard Harroch and Lou Krieger) and recently he published Poker, Life and Other Confusing Things, a collection of essays. In 2012 he proposed a novel framework for the notion "gambling" based on the two dimensions of expected value of a game and the flexibility that a game affords each player.Reber, A. S. (2012). The EVF Model of Gambling: A novel framework for understanding gambling and, by extension, poker.
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The corresponding randomized algorithm is based on the model of boson sampling and it uses the tools proper to quantum optics, to represent the permanent of positive-semidefinite matrices as the expected value of a specific random variable. The latter is then approximated by its sample mean. This algorithm, for a certain set of positive-semidefinite matrices, approximates their permanent in polynomial time up to an additive error, which is more reliable than that of the standard classical polynomial-time algorithm by Gurvits.
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The lower troposphere trend derived from UAH satellites (+0.128 °C/decade) is currently lower than both the GISS and Hadley Centre surface station network trends (+0.161 and +0.160 °C/decade respectively), while the RSS trend (+0.158 °C/decade) is similar. However, if the expected trend in the lower troposphere is indeed higher than the surface, then given the surface data, the troposphere trend would be around 0.194 °C/decade, making the UAH and RSS trends 66% and 81% of the expected value respectively.
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It is suspected of being a Delta Scuti variable star. The companion is a spectral class M0.5V red dwarf star with absolute magnitude 9.80. It is a UV Ceti variable star that undergoes random increases in luminosity. This star is currently separated from the primary by an angle of 6.6 arcseconds, which indicates an orbit with a semimajor axis whose expected value is 206 AU. Alpha Caeli is approximately 65.7 light years from Earth and is an estimated 900 million years old.
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If one is using the frequentist notion of probability, where probabilities are considered to be fixed values, then applying expected value and expected utility to decision-making requires knowing the probabilities of various outcomes. However, in practice there will be many situations where the probabilities are unknown, and one is operating under uncertainty. In economics, Knightian uncertainty or ambiguity may occur. Thus one must make assumptions about the probabilities, but then the expected values of various decisions can be very sensitive to the assumptions.
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The base distribution is the expected value of the process, i.e., the Dirichlet process draws distributions "around" the base distribution the way a normal distribution draws real numbers around its mean. However, even if the base distribution is continuous, the distributions drawn from the Dirichlet process are almost surely discrete. The scaling parameter specifies how strong this discretization is: in the limit of \alpha\rightarrow 0, the realizations are all concentrated at a single value, while in the limit of \alpha\rightarrow\infty the realizations become continuous.
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G. Toussaint profile at McGill University He was a co-founder of the Annual ACM Symposium on Computational Geometry, and the annual Canadian Conference on Computational Geometry. Along with Selim Akl, he was an author and namesake of the efficient "Akl–Toussaint algorithm" for the construction of the convex hull of a planar point set. This algorithm exhibits a computational complexity with expected value linear in the size of the input.Selim G. Akl and Godfried T. Toussaint, "A fast convex hull algorithm," Information Processing Letters, Vol.
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Economist Fischer Black gave the following illustration in 1995. Suppose that the exchange rate between an "apple" country where consumers prefer apples, and an "orange" country where consumers prefer valuable oranges, is currently 1:1, but will change next year to 2:1 or 1:2 with equal probability. Suppose an apple consumer trades an apple to an orange consumer in exchange for an orange. The apple consumer now has given up an apple for an orange, which next year has an expected value of 1.25 apples.
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As above, where the value of an asset in the future is known (or expected), this value can be used to determine the asset's rational price today. In an option contract, however, exercise is dependent on the price of the underlying, and hence payment is uncertain. Option pricing models therefore include logic that either "locks in" or "infers" this future value; both approaches deliver identical results. Methods that lock-in future cash flows assume arbitrage free pricing, and those that infer expected value assume risk neutral valuation.
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For example, a hypothetical population might include 10 million men and 10 million women. Suppose that a biased sample of 100 patients included 20 men and 80 women. A researcher could correct for this imbalance by attaching a weight of 2.5 for each male and 0.625 for each female. This would adjust any estimates to achieve the same expected value as a sample that included exactly 50 men and 50 women, unless men and women differed in their likelihood of taking part in the survey.
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An observed value of 81 would use the tick mark above 81 in range E, and curved scale E would be used for the expected value. This allows five different nomograms to be incorporated into a single diagram. In this manner, the blue line demonstrates the computation of :(9 − 5)2/ 5 = 3.2 and the red line demonstrates the computation of :(81 − 70)2 / 70 = 1.7 In performing the test, Yates's correction for continuity is often applied, and simply involves subtracting 0.5 from the observed values.
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For instance, in the three-dimensional case, where two variables X and Y are given as coordinates, the elevation function between any two points (x1, y1) and (x2, y2) can be set to have a mean or expected value that increases as the vector distance between (x1, y1) and (x2, y2). There are, however, many ways of defining the elevation function. For instance, the fractional Brownian motion variable may be used, or various rotation functions may be used to achieve more natural looking surfaces.
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For example, a trader might buy cheap insurance contracts against a rare but catastrophic risk. The vast majority of the time – and for many years running – the trader will make a small annual loss (the CDS premium) even if the trade has positive expected value. When the rare event occurs, the trader may suddenly have a huge windfall "profit" claim against whoever wrote the "insurance". And this would mean a sudden increase in the relevance of whether or not the 'insurance writing' counterparty can actually pay.
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Bernoulli wrote the text between 1684 and 1689, including the work of mathematicians such as Christiaan Huygens, Gerolamo Cardano, Pierre de Fermat, and Blaise Pascal. He incorporated fundamental combinatorial topics such as his theory of permutations and combinations (the aforementioned problems from the twelvefold way) as well as those more distantly connected to the burgeoning subject: the derivation and properties of the eponymous Bernoulli numbers, for instance. Core topics from probability, such as expected value, were also a significant portion of this important work.
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In this tradition, Medici the board game is based on the pricing of risk: each lot of commodities has an uncertain future value based on how cards are drawn from the deck, what other players buy, and other factors. In order to play the game well, players must judge and price the risk attached to each lot of cards, buying them for a price appropriate to their expected value and the riskiness of the investment. Medici placed 5th in the 1995 Deutscher Spiele Preis.
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In quantum mechanics, the expectation value is the probabilistic expected value of the result (measurement) of an experiment. It can be thought of as an average of all the possible outcomes of a measurement as weighted by their likelihood, and as such it is not the most probable value of a measurement; indeed the expectation value may have zero probability of occurring (e.g. measurements which can only yield integer values may have a non-integer mean). It is a fundamental concept in all areas of quantum physics.
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If not, one can just do 213 more trials and hope for a total of 226 successes; if not, just repeat as necessary. Lazzarini performed 3408 = 213 · 16 trials, making it seem likely that this is the strategy he used to obtain his "estimate." The above description of strategy might even be considered charitable to Lazzarini. A statistical analysis of intermediate results he reported for fewer tosses leads to a very low probability of achieving such close agreement to the expected value all through the experiment.
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19:5 A similar technique is used in the probability model of credit default swap (CDS) valuation. rNPV modifies the standard NPV calculation of discounted cash flow (DCF) analysis by adjusting (multiplying) each cash flow by the estimated probability that it occurs (the estimated success rate). In the language of probability theory, the rNPV is the expected value. Note that this in contrast to the more general valuation approach, where risk is instead incorporated by adding a risk premium percentage to the discount rate, i.e.
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A mean-preserving spread (MPS) is a change from one probability distribution A to another probability distribution B, where B is formed by spreading out one or more portions of A's probability density function while leaving the mean (the expected value) unchanged. The concept of a mean-preserving spread provides a partial ordering of probability distributions according to their dispersions: of two probability distributions, one may be ranked as having more dispersion than the other, or alternatively neither may be ranked as having more dispersion.
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In plain language, it is meaningful to say that 20 Kelvin is twice as hot as 10 Kelvin, but only in this scale with a true absolute zero. While a standard deviation (SD) can be measured in Kelvin, Celsius, or Fahrenheit, the value computed is only applicable to that scale. Only the Kelvin scale can be used to compute a valid coefficient of variability. Measurements that are log-normally distributed exhibit stationary CV; in contrast, SD varies depending upon the expected value of measurements.
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In addition to his work in cognitive psychology and the philosophy of mind, Reber has had a parallel career as a reporter and commentator on gambling, particularly poker. As a free-lance writer, he has authored hundreds of columns, most from the psychologist’s point of view. These have been published in magazines such as Casino Player, Strictly Slots and Poker Pro Magazine and web sites like PokerListings.com. His breakdown of forms of gambling based on expected value was presented in The New Gambler’s Bible.
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In decision theory, the expected value of sample information (EVSI) is the expected increase in utility that a decision-maker could obtain from gaining access to a sample of additional observations before making a decision. The additional information obtained from the sample may allow them to make a more informed, and thus better, decision, thus resulting in an increase in expected utility. EVSI attempts to estimate what this improvement would be before seeing actual sample data; hence, EVSI is a form of what is known as preposterior analysis.
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However, it is known from experiments that new physics such as superpartners does not occur at very low energy scales, so even if these new particles reduce the loop corrections, they do not reduce them enough to make the renormalized Higgs mass completely natural. The expected value of the Higgs mass is about 10 percent of the size of the loop corrections which shows that a certain "little" amount of fine-tuning seems necessary. Particle physicists have different opinions as to whether the little hierarchy problem is serious.
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An example of a problem which causes difficulty and debate is the St. Petersburg paradox. This is a lottery which is constructed so that the expected value is infinite but unlikely so that most people will not pay a large fee to play. Gerd Gigerenzer explained that, in this case, mathematicians refined their formulae to model this pragmatic behaviour. Keith Stanovich characterizes this as a Panglossian position in the debate—that humans are fundamentally rational and any variance between the normative position and empirical outcomes may be explained by such adjustments.
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Ordinary linear regression predicts the expected value of a given unknown quantity (the response variable, a random variable) as a linear combination of a set of observed values (predictors). This implies that a constant change in a predictor leads to a constant change in the response variable (i.e. a linear-response model). This is appropriate when the response variable can vary, to a good approximation, indefinitely in either direction, or more generally for any quantity that only varies by a relatively small amount compared to the variation in the predictive variables, e.g.
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Example of the folded cumulative distribution for a normal distribution function with an expected value of 0 and a standard deviation of 1. While the plot of a cumulative distribution often has an S-like shape, an alternative illustration is the folded cumulative distribution or mountain plot, which folds the top half of the graph over, thus using two scales, one for the upslope and another for the downslope. This form of illustration emphasises the median and dispersion (specifically, the mean absolute deviation from the median) of the distribution or of the empirical results.
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Choice under uncertainty is often characterized as the maximization of expected utility. Utility is often assumed to be a function of profit or final portfolio wealth, with a positive first derivative. The utility function whose expected value is maximized is concave for a risk averse agent, convex for a risk lover, and linear for a risk neutral agent. Thus in the risk neutral case, expected utility of wealth is simply equal to the expectation of a linear function of wealth, and maximizing it is equivalent to maximizing expected wealth itself.
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Market measures which assign probabilities to financial market spaces based on actual market movements are examples of probability measures which are of interest in mathematical finance, e.g. in the pricing of financial derivatives.Quantitative methods in derivatives pricing by Domingo Tavella 2002 page 11 For instance, a risk-neutral measure is a probability measure which assumes that the current value of assets is the expected value of the future payoff taken with respect to that same risk neutral measure (i.e. calculated using the corresponding risk neutral density function), and discounted at the risk-free rate.
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Intertemporal portfolio choice is the process of allocating one's investable wealth to various assets, especially financial assets, repeatedly over time, in such a way as to optimize some criterion. The set of asset proportions at any time defines a portfolio. Since the returns on almost all assets are not fully predictable, the criterion has to take financial risk into account. Typically the criterion is the expected value of some concave function of the value of the portfolio after a certain number of time periods--that is, the expected utility of final wealth.
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The Kelly criterion for intertemporal portfolio choice states that, when asset return distributions are identical in all periods, a particular portfolio replicated each period will outperform all other portfolio sequences in the long run. Here the long run is an arbitrarily large number of time periods such that the distributions of observed outcomes for all assets match their ex ante probability distributions. The Kelly criterion gives rise to the same portfolio decisions as does the maximization of the expected value of the log utility function as described above.
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Thus, even though the expected value of sin x1 for the source test case x1 = 1.234 correct to the required accuracy is not known, a follow-up test case x2 = π − 1.234 can be constructed. We can verify whether the actual outputs produced by the program under test from the source test case and the follow-up test case are consistent with the MR in question. Any inconsistency (after taking rounding errors into consideration) indicates a failure of the implementation. MRs are not limited to programs with numerical inputs or equality relations.
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We consider a European option (say, a call) on the forward F strike at K, which expires T years from now. The value of this option is equal to the suitably discounted expected value of the payoff \max(F_T-K,\;0) under the probability distribution of the process F_t. Except for the special cases of \beta=0 and \beta=1, no closed form expression for this probability distribution is known. The general case can be solved approximately by means of an asymptotic expansion in the parameter \varepsilon=T\alpha^2.
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Game square (that is, the payoff to player I) for a game with no value, due to Sion and Wolfe. The payoff is 0 along the two diagonal lines In the mathematical theory of games, in particular the study of zero-sum continuous games, not every game has a minimax value. This is the expected value to one of the players when both play a perfect strategy (which is to choose from a particular PDF). This article gives an example of a zero-sum game that has no value.
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A mark is made indicating how far the actual result was from the mean, which is the expected value for the control. Lines run across the graph at the mean, as well as one, two and three standard deviations to either side of the mean. This makes it easy to see how far off the result was. Rules such as the Westgard rules can be applied to see whether the results from the samples when the control was done can be released, or if they need to be rerun.
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The triangular distribution is also commonly used. It differs from the double-triangular by its simple triangular shape and by the property that the mode does not have to coincide with the median. The mean (expected value) is then: : E = (a + m + b) / 3. In some applications,Ministry of Defence (2007) "Three point estimates and quantitative risk analysis" Policy, information and guidance on the Risk Management aspects of UK MOD Defence Acquisition the triangular distribution is used directly as an estimated probability distribution, rather than for the derivation of estimated statistics.
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100-yr, 50-yr, 25-yr, and so forth), and n is the number of years in the period. The probability of exceedance Pe is also described as the natural, inherent, or hydrologic risk of failure.Mays, L.W (2005) Water Resources Engineering, chapter 10, Probability, risk, and uncertainty analysis for hydrologic and hydraulic design Hoboken: J. Wiley & SonsMaidment, D.R. ed.(1993) Handbook of Hydrology, chapter 18, Frequency analysis of extreme events New York: McGraw-Hill However, the expected value of the number of 100-year floods occurring in any 100-year period is 1.
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Prices of assets depend crucially on their risk as investors typically demand more profit for bearing more risk. Therefore, today's price of a claim on a risky amount realised tomorrow will generally differ from its expected value. Most commonly, investors are risk-averse and today's price is below the expectation, remunerating those who bear the risk (at least in large financial markets; examples of risk-seeking markets are casinos and lotteries). To price assets, consequently, the calculated expected values need to be adjusted for an investor's risk preferences (see also Sharpe ratio).
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The simulated random numbers originate from a bivariate normal distribution with a variance of 1 and a deviation of the expected value of 0.4. The significance level is 5% and the number of cases is 60. Two-sample t-tests for a difference in mean involve independent samples (unpaired samples) or paired samples. Paired t-tests are a form of blocking, and have greater power than unpaired tests when the paired units are similar with respect to "noise factors" that are independent of membership in the two groups being compared.
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The PD game is not the only similarity sensitive game. Games for which the choice of the action with the higher expected value depends on the value of ps are defined as Similarity Sensitive Games (SSGs), whereas others are nonsimilarity sensitive. Focusing only on the 24 completely rank-ordered and symmetric games, we can mark 12 SSGs. After eliminating games that reflect permutations of other games generated either by switching rows, columns, or both rows and columns, we are left with six basic (completely rank-ordered and symmetric) SSGs.
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Although the subject of sexual dimorphism is not in itself controversial, the measures by which it is assessed differ widely. Most of the measures are used on the assumption that a random variable is considered so that probability distributions should be taken into account. In this review, a series of sexual dimorphism measures are discussed concerning both their definition and the probability law on which they are based. Most of them are sample functions, or statistics, which account for only partial characteristics, for example the mean or expected value, of the distribution involved.
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56 Probabilistic research of expected value scenarios shows that by splitting eights one can convert a hand that presents an expected loss to two hands that may present an expected profit or a reduced loss, depending on what the dealer is showing.Hagen and Wiess, pp. 66-67 A split pair of eights is expected to win against dealer upcards of 2 through 7 and to lose less against dealer upcards of 8 through ace. If a player hits on a pair of eights, he is expected to lose $52 for a $100 bet.
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The Shannon information is closely related to information theoretic entropy, which is the expected value of the self-information of a random variable, quantifying how surprising the random variable is "on average." This is the average amount of self-information an observer would expect to gain about a random variable when measuring it.Jones, D.S., Elementary Information Theory, Vol., Clarendon Press, Oxford pp 11-15 1979 The information content can be expressed in various units of information, of which the most common is the "bit" (sometimes also called the "shannon"), as explained below.
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A related functional that shares the translation-invariance and homogeneity properties with the nth central moment, but continues to have this additivity property even when n ≥ 4 is the nth cumulant κn(X). For n = 1, the nth cumulant is just the expected value; for n = either 2 or 3, the nth cumulant is just the nth central moment; for n ≥ 4, the nth cumulant is an nth-degree monic polynomial in the first n moments (about zero), and is also a (simpler) nth-degree polynomial in the first n central moments.
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The workmanship is analysed in this work. The authors emphasise the importance of thermal bridges that were not considered for the calculations, and how those originated by the internal partitions that separate dwellings have the largest impact on the final energy use. The dwellings that were monitored in use in this study show a large difference between the real energy use and that estimated using SAP, with one of them giving +176% of the expected value when in use. Hopfe has published several papers concerning uncertainties in building design that cover workmanship.
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The use of mean squared error without question has been criticized by the decision theorist James Berger. Mean squared error is the negative of the expected value of one specific utility function, the quadratic utility function, which may not be the appropriate utility function to use under a given set of circumstances. There are, however, some scenarios where mean squared error can serve as a good approximation to a loss function occurring naturally in an application. Like variance, mean squared error has the disadvantage of heavily weighting outliers.
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In probability theory, Hoeffding's inequality provides an upper bound on the probability that the sum of bounded independent random variables deviates from its expected value by more than a certain amount. Hoeffding's inequality was proven by Wassily Hoeffding in 1963. Hoeffding's inequality is a generalization of the Chernoff bound, which applies only to Bernoulli random variables; for a more intuitive proof, see this note, and a special case of the Azuma–Hoeffding inequality and the McDiarmid's inequality. It is similar to, but incomparable with, the Bernstein inequality, proved by Sergei Bernstein in 1923.
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Surrendering has a slightly higher advantage for the house in the case that a bonus payout is offered, so based on the expected value probabilities a player should never surrender. The dealer and the player each have a 46.3% chance of winning on the first card (in a standard game with 6 decks), so this seems like an even money game. The house advantage, however, comes from what happens in the case of a tie. The house advantage increases with the number of decks in play and decreases in casinos who offer a bonus payout.
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Choice under uncertainty is often characterized as the maximization of expected utility. Utility is often assumed to be a function of profit or final portfolio wealth, with a positive first derivative. The utility function whose expected value is maximized is convex for a risk- seeker, concave for a risk-averse agent, and linear for a risk-neutral agent. Its convexity in the risk-seeking case has the effect of causing a mean- preserving spread of any probability distribution of wealth outcomes to be preferred over the unspread distribution.
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Parlay bets are paid out at odds higher than the typical single game bet, but still below the "true" odds. For instance, a common 2-team NFL parlay based entirely on the spread generally has a payout of 2.6:1. In reality however, if one assumes that each single game bet is 50/50, the true payout should instead be 3:1 (10% expected value for the house). A house may average 20-30% profit on spread parlays compared to perhaps 4.5% profit on individual sports mix parlay bets.
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Switch 1 (S1) and switch 2 (S2) are used to enable different measurement steps. Detailed process flow of this method, including an experiment part and a data analysis part. First of all, it is critical to keep the surface charge density identical as reflected by QSC,max, to ensure the consistency of measurement at different x. Thus in Step 1, S1 was turned on and S2 was turn off to measure QSC,max; if QSC,max is lower than the expected value, additional triboelectrification process is conducted to approach that.
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Bid shading is also used in first-price auctions, where the winning bidder pays the amount of his bid. If a participant bids an amount equal to their value for the good, they would gain nothing by winning the auction, since they are indifferent between the money and the good. Bidders will optimize their expected value by accepting a lower chance of winning in return for a higher payoff if they win. In a first- price common value auction, a savvy bidder should shade for both of the above purposes.
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Not limited to real-valued random variables and linear dependence like the correlation coefficient, MI is more general and determines how different the joint distribution of the pair (X,Y) is to the product of the marginal distributions of X and Y. MI is the expected value of the pointwise mutual information (PMI). The quantity was defined and analyzed by Claude Shannon in his landmark paper A Mathematical Theory of Communication, although he did not call it "mutual information". This term was coined later by Robert Fano. Mutual Information is also known as information gain.
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No matter what the initial state, the cat will eventually catch the mouse (with probability 1) and a stationary state π = (0,0,0,0,1) is approached as a limit. To compute the long-term average or expected value of a stochastic variable Y, for each state Sj and time tk there is a contribution of Yj,k·P(S=Sj,t=tk). Survival can be treated as a binary variable with Y=1 for a surviving state and Y=0 for the terminated state. The states with Y=0 do not contribute to the long-term average.
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The workmanship is analysed in this work. The authors emphasise the importance of thermal bridges that were not considered for the calculations, and how those originated by the internal partitions that separate dwellings have the largest impact on the final energy use. The dwellings that were monitored in use in this study show a large difference between the real energy use and that estimated using SAP, with one of them giving +176% of the expected value when in use. Hopfe has published several papers concerning uncertainties in building design that cover workmanship.
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This estimator can be interpreted as a weighted average between the noisy measurements y and the prior expected value m. If the noise variance \sigma_w^2 is low compared with the variance of the prior \sigma_x^2 (corresponding to a high SNR), then most of the weight is given to the measurements y, which are deemed more reliable than the prior information. Conversely, if the noise variance is relatively higher, then the estimate will be close to m, as the measurements are not reliable enough to outweigh the prior information.
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Generally the conductivity of a solution increases with temperature, as the mobility of the ions increases. For comparison purposes reference values are reported at an agreed temperature, usually 298 K (≈ 25 °C), although occasionally 20 °C is used. So called 'compensated' measurements are made at a convenient temperature but the value reported is a calculated value of the expected value of conductivity of the solution, as if it had been measured at the reference temperature. Basic compensation is normally done by assuming a linear increase of conductivity versus temperature of typically 2% per Kelvin.
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2018 estimates for objects at least 1 km in size put the figure somewhere between 89% and 99%, with an expected value of 94%. This matches the figure from a 2017 NASA report which was estimated independently using a different technique The effectiveness of the catalogue is somewhat limited by the fact that some proportion of the objects have been lost since their discovery. Their orbits were not determined sufficiently well to allow their future position to be predicted with accuracy, so we no longer know what part of their orbit that they are on.
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From relatively early on, it was accepted that some of these conditions would be violated by real decision-makers in practice but that the conditions could be interpreted nonetheless as 'axioms' of rational choice. Until the mid-twentieth century, the standard term for the expected utility was the moral expectation, contrasted with "mathematical expectation" for the expected value."Moral expectation", under Jeff Miller, Earliest Known Uses of Some of the Words of Mathematics (M) , accessed 2011-03-24. The term "utility" was first introduced mathematically in this connection by Jevons in 1871; previously the term "moral value" was used.
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Note that W, and consequently its infinitesimal increment dW, represents the only source of uncertainty in the price history of the stock. Intuitively, W(t) is a process that "wiggles up and down" in such a random way that its expected change over any time interval is 0. (In addition, its variance over time T is equal to T; see ); a good discrete analogue for W is a simple random walk. Thus the above equation states that the infinitesimal rate of return on the stock has an expected value of μ dt and a variance of \sigma^2 dt .
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It had been naively believed that the quark sea in the proton was formed by quantum chromodynamics (QCD) processes that did not discriminate between up and down quarks. However, results of deep inelastic scattering of high energy muons on a proton and a deuteron targets by CERN-NMC showed that there are more 's than 's in the proton. The Gottfried sum measured by NMC was 0.235±0.026, which is significantly smaller than the expected value of 1/3. This means that (x)-(x) integrated over Bjorken x from 0 to 1.0 is 0.147±0.039, indicating a flavor asymmetry in the proton sea.
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The approximate nature of Maxwell's equations becomes more and more apparent when going into the extremely strong field regime (see Euler–Heisenberg Lagrangian) or to extremely small distances. Finally, Maxwell's equations cannot explain any phenomenon involving individual photons interacting with quantum matter, such as the photoelectric effect, Planck's law, the Duane–Hunt law, and single-photon light detectors. However, many such phenomena may be approximated using a halfway theory of quantum matter coupled to a classical electromagnetic field, either as external field or with the expected value of the charge current and density on the right hand side of Maxwell's equations.
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FFW programs are often used to counter a perceived dependency syndrome associated with freely distributed food. However, poorly designed FFW programs may cause more risk of harming local production than the benefits of free food distribution. In structurally weak economies, FFW program design is not as simple as determining the appropriate wage rate. Empirical evidence from rural Ethiopia shows that higher-income households had excess labor and thus lower (not higher as expected) value of time, and therefore allocated this labor to FFW schemes in which poorer households could not afford to participate due to labor scarcity.
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Another topic that has seen a thorough development is the theory of uniform distribution mod 1. Take a sequence a1, a2, ... of real numbers and consider their fractional parts. That is, more abstractly, look at the sequence in R/Z, which is a circle. For any interval I on the circle we look at the proportion of the sequence's elements that lie in it, up to some integer N, and compare it to the proportion of the circumference occupied by I. Uniform distribution means that in the limit, as N grows, the proportion of hits on the interval tends to the 'expected' value.
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The rationale for this is that 16 is the square root of 256, which is approximately the number of trading days in a year (252). This also uses the fact that the standard deviation of the sum of n independent variables (with equal standard deviations) is √n times the standard deviation of the individual variables. The average magnitude of the observations is merely an approximation of the standard deviation of the market index. Assuming that the market index daily changes are normally distributed with mean zero and standard deviation σ, the expected value of the magnitude of the observations is √(2/)σ = 0.798σ.
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In probability theory and intertemporal portfolio choice, the Kelly criterion (or Kelly strategy, Kelly bet, ...), also known as the scientific gambling method, is a formula for bet sizing that leads almost surely to higher wealth compared to any other strategy in the long run (i.e. approaching the limit as the number of bets goes to infinity). The Kelly bet size is found by maximizing the expected value of the logarithm of wealth, which is equivalent to maximizing the expected geometric growth rate. The Kelly Criterion is to bet a predetermined fraction of assets, and it can seem counterintuitive.
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Bubbles in financial markets have been studied not only through historical evidence, but also through experiments, mathematical and statistical works. Smith, Suchanek and Williams designed a set of experiments in which an asset that gave a dividend with expected value 24 cents at the end of each of 15 periods (and were subsequently worthless) was traded through a computer network. Classical economics would predict that the asset would start trading near $3.60 (15 times $0.24) and decline by 24 cents each period. They found instead that prices started well below this fundamental value and rose far above the expected return in dividends.
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Since a gambler with infinite wealth will, almost surely, eventually flip heads, the martingale betting strategy was seen as a sure thing by those who advocated it. None of the gamblers possessed infinite wealth, and the exponential growth of the bets would eventually bankrupt "unlucky" gamblers who chose to use the martingale. The gambler usually wins a small net reward, thus appearing to have a sound strategy. However, the gambler's expected value does indeed remain zero (or less than zero) because the small probability that the gambler will suffer a catastrophic loss exactly balances with the expected gain.
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The family of all normal distributions, parametrized by the expected value μ and the variance σ2 ≥ 0, with the Riemannian metric given by the Fisher information matrix, is a statistical manifold. Its geometry is modeled on hyperbolic space. A simple example of a statistical manifold, taken from physics, would be the canonical ensemble: it is a one-dimensional manifold, with the temperature T serving as the coordinate on the manifold. For any fixed temperature T, one has a probability space: so, for a gas of atoms, it would be the probability distribution of the velocities of the atoms.
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The pill jar puzzle is a probability puzzle, which asks the expected value of the number of half-pills remaining when the last whole pill is popped from a jar initially containing whole pills and the way to proceed is by removing a pill from the bottle at random. If the pill removed is a whole pill, it is broken into two half pills. One half pill is consumed and the other one is returned to the jar. If the pill removed is a half pill, then it is simply consumed and nothing is returned to the jar.
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In either case, if any time remains in the half, the team proceeds to a kickoff. Various sources estimate the success rate of a two-point conversion to be between 40% and 55%, significantly lower than that of the extra point, though if the higher value is to be believed, a higher expected value is achieved through the two-point conversion than the extra point.412sportsanalytics "Two-Point Conversion: My two-data-cents,", 2016.K. Pelechrinis "Decision Making in American Football: Evidence from 7 Years of NFL Data, in Machine Learning and Data Mining for Sports Analytics", 2016.
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In late 2007, NOνA passed a Department of Energy "Critical Decision 2" review, meaning roughly that its design, cost, schedule, and scientific goals had been approved. This also allowed the project to be included in the Department of Energy congressional budget request. (NOνA still required a "Critical Decision 3" review to begin construction.) On 21 December 2007, President Bush signed an omnibus spending bill, H.R. 2764, which cut the funding for high energy physics by 88 million dollars from the expected value of 782 million dollars. The budget of Fermilab was cut by 52 million dollars.
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E3) A patient is waiting for a suitable matching kidney donor for a transplant. If the probability that a randomly selected donor is a suitable match is p=0.1, what is the expected number of donors who will be tested before a matching donor is found? With p = 0.1, the mean number of failures before the first success is E(Y) = (1 − p)/p =(1 − 0.1)/0.1 = 9. For the alternative formulation, where X is the number of trials up to and including the first success, the expected value is E(X) = 1/p = 1/0.1 = 10.
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Sometimes a made hand needs to draw to a better hand. For example, if a player has two pair or three of a kind, but an opponent has a straight or flush, to win the player must draw an out to improve to a full house (or four of a kind). There are a multitude of potential situations where one hand needs to improve to beat another, but the expected value of most drawing plays can be calculated by counting outs, computing the probability of winning, and comparing the probability of winning to the pot odds.
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Security breaches continue to cost businesses billions of dollars but a survey revealed that 66% of security staffs do not believe senior leadership takes cyber precautions as a strategic priority. However, reasonable estimates of the financial cost of security breaches can actually help organizations make rational investment decisions. According to the classic Gordon-Loeb Model analyzing the optimal investment level in information security, one can conclude that the amount a firm spends to protect information should generally be only a small fraction of the expected loss (i.e., the expected value of the loss resulting from a cyber/information security breach).
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In accounting and finance, a risk-seeker or risk-lover is a person who has a preference for risk. While most investors are considered risk averse, one could view casino-goers as risk-seeking. If offered either $50 or a 50% each chance of either $100 or nothing, a risk-seeking person would prefer the gamble even though the gamble and the sure thing have the same expected value. Risk-seeking behavior can be observed in the negative domain x<0 for prospect theory value functions, where the functions are convex for x<0 but concave for x > 0.
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Company researchers gathered detailed medical information on each client, using this as the basis for the creation of personalized research reports for various conditions (or, in some cases, for the purpose of client performance enhancement). It also assessed the expected value of various tests, and created maps of correlations between possible medical conditions. One aim of the company was to aid doctors with advanced artificial intelligence and data from information experts. MetaMed's personalized medical research services were targeted at the market for concierge medicine, with prices ranging from a few thousand dollars to hundreds of thousands.
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Thorium's boiling point of 4788 °C is the fifth-highest among all the elements with known boiling points. The properties of thorium vary widely depending on the degree of impurities in the sample. The major impurity is usually thorium dioxide (ThO2); even the purest thorium specimens usually contain about a tenth of a percent of the dioxide. Experimental measurements of its density give values between 11.5 and 11.66 g/cm3: these are slightly lower than the theoretically expected value of 11.7 g/cm3 calculated from thorium's lattice parameters, perhaps due to microscopic voids forming in the metal when it is cast.
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More generally, the expected value of a function of a random variable is the probability-weighted average of the values the function takes on for each possible value of the random variable. In regressions in which the dependent variable is assumed to be affected by both current and lagged (past) values of the independent variable, a distributed lag function is estimated, this function being a weighted average of the current and various lagged independent variable values. Similarly, a moving average model specifies an evolving variable as a weighted average of current and various lagged values of a random variable.
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In statistics, Poisson regression is a generalized linear model form of regression analysis used to model count data and contingency tables. Poisson regression assumes the response variable Y has a Poisson distribution, and assumes the logarithm of its expected value can be modeled by a linear combination of unknown parameters. A Poisson regression model is sometimes known as a log-linear model, especially when used to model contingency tables. Negative binomial regression is a popular generalization of Poisson regression because it loosens the highly restrictive assumption that the variance is equal to the mean made by the Poisson model.
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The triangle demonstrates many mathematical properties in addition to showing binomial coefficients. In 1654, prompted by his friend the Chevalier de Méré, he corresponded with Pierre de Fermat on the subject of gambling problems, and from that collaboration was born the mathematical theory of probabilities. The specific problem was that of two players who want to finish a game early and, given the current circumstances of the game, want to divide the stakes fairly, based on the chance each has of winning the game from that point. From this discussion, the notion of expected value was introduced.
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Autocovariance can be used to calculate turbulent diffusivity. Turbulence in a flow can cause the fluctuation of velocity in space and time. Thus, we are able to identify turbulence through the statistics of those fluctuations. Reynolds decomposition is used to define the velocity fluctuations u'(x,t) (assume we are now working with 1D problem and U(x,t) is the velocity along x direction): :U(x,t) = \langle U(x,t) \rangle + u'(x,t), where U(x,t) is the true velocity, and \langle U(x,t) \rangle is the expected value of velocity.
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Q-learning is a model-free reinforcement learning algorithm to learn quality of actions telling an agent what action to take under what circumstances. It does not require a model (hence the connotation "model-free") of the environment, and it can handle problems with stochastic transitions and rewards, without requiring adaptations. For any finite Markov decision process (FMDP), Q-learning finds an optimal policy in the sense of maximizing the expected value of the total reward over any and all successive steps, starting from the current state. Q-learning can identify an optimal action-selection policy for any given FMDP, given infinite exploration time and a partly-random policy.
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As a quantitative measure, the "forecast bias" can be specified as a probabilistic or statistical property of the forecast error. A typical measure of bias of forecasting procedure is the arithmetic mean or expected value of the forecast errors, but other measures of bias are possible. For example, a median- unbiased forecast would be one where half of the forecasts are too low and half too high: see Bias of an estimator. In contexts where forecasts are being produced on a repetitive basis, the performance of the forecasting system may be monitored using a tracking signal, which provides an automatically maintained summary of the forecasts produced up to any given time.
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In essence, the compensation scheme becomes more like a call option on performance (which increases in value with increased volatility (cf. options pricing). If you are one of ten players competing for the asymmetrically large top prize, you may benefit from reducing the expected value of your overall performance to the firm in order to increase your chance that you have an outstanding performance (and win the prize). In moderation this can offset the greater risk aversion of agents vs principals because their social capital is concentrated in their employer while in the case of public companies the principal typically owns his stake as part of a diversified portfolio.
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Many standard estimators can be improved, in terms of mean squared error (MSE), by shrinking them towards zero (or any other fixed constant value). In other words, the improvement in the estimate from the corresponding reduction in the width of the confidence interval can outweigh the worsening of the estimate introduced by biasing the estimate towards zero (see bias-variance tradeoff). Assume that the expected value of the raw estimate is not zero and consider other estimators obtained by multiplying the raw estimate by a certain parameter. A value for this parameter can be specified so as to minimize the MSE of the new estimate.
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Underhill, L.G.; Bradfield d. (1998) Introstat, Juta and Company Ltd. p. 181 In probability and statistics, the population mean, or expected value, is a measure of the central tendency either of a probability distribution or of the random variable characterized by that distribution. In a discrete probability distribution of a random variable X, the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value x of X and its probability p(x), and then adding all these products together, giving \mu = \sum x p(x).
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The Fundamental Theorem of Poker applies to all heads-up decisions, but it does not apply to all multi-way decisions. This is because each opponent of a player can make an incorrect decision, but the "collective decision" of all the opponents works against the player. This type of situation occurs mostly in games with multi-way pots, when a player has a strong hand, but several opponents are chasing with draws or other weaker hands. Also, a good example is a player with a deep stack making a play that favors a short-stacked opponent because he can extract more expected value from the other deep-stacked opponents.
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One could apply Pearson's chi-square test of whether the population distribution is a Poisson distribution with expected value 3.3. However, the null hypothesis did not specify that it was that particular Poisson distribution, but only that it is some Poisson distribution, and the number 3.3 came from the data, not from the null hypothesis. A rule of thumb says that when a parameter is estimated, one reduces the number of degrees of freedom by 1, in this case from 9 (since there are 10 cells) to 8\. One might hope that the resulting test statistic would have approximately a chi-square distribution when the null hypothesis is true.
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The methods include consideration of the solving of streams of problems in environments over time. In related work, he applied probability and machine learning to identify hard problems and to guide theorem proving. He introduced the anytime algorithm paradigm in AI, where partial results, probabilities, or utilities of outcomes are refined with computation under different availabilities or costs of time, guided by the expected value of computation. He has issued long-term challenge problems for AI—and has espoused a vision of open-world AI, where machine intelligences have the ability to understand and perform well in the larger world where they encounter situations they have not seen before.
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Most consumers care about the nationality only insofar as it influences which wine is trendy. The only exceptions are a single loyalist American consumer, who only drinks American wine and is indifferent to trendiness, and a single loyalist German consumer, whose only drinks German wine and is likewise indifferent to trendiness. The American loyalist counter-intuitively prefers to hold a German rather than an American wine in November, as it has a 50% chance of being tradeable for 0.5 American wines, and a 50% chance of being tradeable for 2 American wines, and thus has an expected value of 1.25 American wines. (All the consumers in this scenario are risk-neutral).
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The main state variable of the model is the short rate, which is assumed to follow the stochastic differential equation (under the risk-neutral measure): : d\ln(r) = [\theta_t-\phi_t \ln(r)] \, dt + \sigma_t\, dW_t where dWt is a standard Brownian motion. The model implies a log-normal distribution for the short rate and therefore the expected value of the money-market account is infinite for any maturity. In the original article by Fischer Black and Piotr Karasinski the model was implemented using a binomial tree with variable spacing, but a trinomial tree implementation is more common in practice, typically a lognormal application of the Hull-White Lattice.
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In decision theory, the von Neumann–Morgenstern (or VNM) utility theorem shows that, under certain axioms of rational behavior, a decision-maker faced with risky (probabilistic) outcomes of different choices will behave as if he or she is maximizing the expected value of some function defined over the potential outcomes at some specified point in the future. This function is known as the von Neumann–Morgenstern utility function. The theorem is the basis for expected utility theory. In 1947, John von Neumann and Oskar Morgenstern proved that any individual whose preferences satisfied four axioms has a utility function;Neumann, John von and Morgenstern, Oskar, Theory of Games and Economic Behavior.
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Princeton, NJ. Princeton University Press, 1953. such an individual's preferences can be represented on an interval scale and the individual will always prefer actions that maximize expected utility. That is, they proved that an agent is (VNM-)rational if and only if there exists a real-valued function u defined by possible outcomes such that every preference of the agent is characterized by maximizing the expected value of u, which can then be defined as the agent's VNM-utility (it is unique up to adding a constant and multiplying by a positive scalar). No claim is made that the agent has a "conscious desire" to maximize u, only that u exists.
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There are several theories concerning the algorithm that “The Bank” uses to determine the appropriate bank offer. This is a secret held by the various publishers around the world, however a number of people have approximated the algorithm with various levels of accuracy. In many variations of the format the Bank does not know the contents of the briefcase, and therefore the Monty Hall Problem does not apply to the probability calculations, but this varies from country to country. Statistical studies of the US version of the show were undertaken by Daniel Shifflet in 2011, and showed a linear regression of bank offers against expected value.
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In summary, Shifflet found that the bank would offer a percentage of the expected value (EV) of the remaining cases, and this percentage increased linearly from approximately 37% of EV at the first offer to approximately 84% of EV at the seventh offer. This version of the program also allowed players to ‘hypothetically’ play out the remainder of the game from the point where they accepted the bank's offer, and Shiffler noted that the hypothetical bank offers were significantly higher than real bank offers at equivalent points in the game. Keep in mind, that this is for the syndicated 30-minute version of the show.
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The house edge (HE) or vigorish is defined as the casino profit expressed as a percentage of the player's original bet. In games such as Blackjack or Spanish 21, the final bet may be several times the original bet, if the player doubles or splits. Example: In American Roulette, there are two zeroes and 36 non-zero numbers (18 red and 18 black). If a player bets $1 on red, his chance of winning $1 is therefore 18/38 and his chance of losing $1 (or winning -$1) is 20/38. The player's expected value, EV = (18/38 x 1) + (20/38 x -1) = 18/38 - 20/38 = -2/38 = -5.26%.
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Weaver and Frederick (2012) presented their participants with retail prices of products, and then asked them to specify either their buying or selling price for these products. The results revealed that sellers’ valuations were closer to the known retail prices than those of buyers. A second line of studies is a meta-analysis of buying and selling of lotteries. A review of over 30 empirical studies showed that selling prices were closer to the lottery's expected value, which is the normative price of the lottery: hence the endowment effect was consistent with buyers’ tendency to under-price lotteries as compared to the normative price.
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Unicity (\varepsilon_p) is formally defined as the expected value of the fraction of uniquely identifiable trajectories, given p points selected from those trajectories uniformly at random. A full computation of \varepsilon_p of a data set D requires picking p points uniformly at random from each trajectory T_i \in D, and then checking whether or not any other trajectory also contains those p points. Averaging over all possible sets of p points for each trajectory results in a value for \varepsilon_p. This is usually prohibitively expensive as it requires considering every possible set of p points for each trajectory in the data set — trajectories that sometimes contain thousands of points.
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The coaches' choice of whether to attempt a one- or two-point conversion depends on the game's current score, the amount of time remaining, and their assessment of their team's chance of success. Analysis of historical data finds that the two-point conversion is successful about half the time, whereas one-point kicks are almost always successful. Therefore the expected value of both options is roughly similar, with the critical factor being whether the chance of a successful two-point conversion is more or less than half that of a successful kick. However, the mathematics regarding maximizing a team's chances of winning are more complicated.
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Expected Utility Theory (EUT) poses a utility calculation linearly combining weights and values of the probabilities associated with various outcomes. By presuming that decision-makers themselves incorporate an accurate weighting of probabilities into calculating expected values for their decision-making, EUT assumes that people's subjective probability-weighting matches objective probability differences, when they are, in reality, exceedingly disparate. Consider the choice between a prospect that offers an 85% chance to win $1000 (with a 15% chance to win nothing) and the alternative of receiving $800 for sure. A large majority of people prefer the sure thing over the gamble, although the gamble has higher (mathematical) expected value (also known as expectation).
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It follows from concavity that the subjective value attached to a gain of $800 is more than 80% of the value of a gain of $1,000. Consequently, the concavity of the utility function entails a risk averse preference for a sure gain of $800 over an 80% chance to win $1,000, although the two prospects have the same monetary expected value. While EUT has dominated the analysis of decision-making under risk and has generally been accepted as a normative model of rational choice (telling us how we should make decisions), descriptive models of how people actually behave deviate significantly from this normative model.
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In mathematical statistics, the Fisher information (sometimes simply called informationLehmann & Casella, p. 115) is a way of measuring the amount of information that an observable random variable X carries about an unknown parameter θ of a distribution that models X. Formally, it is the variance of the score, or the expected value of the observed information. In Bayesian statistics, the asymptotic distribution of the posterior mode depends on the Fisher information and not on the prior (according to the Bernstein–von Mises theorem, which was anticipated by Laplace for exponential families). Lucien Le Cam (1986) Asymptotic Methods in Statistical Decision Theory: Pages 336 and 618–621 (von Mises and Bernstein).
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In addition to using different universalization conditions, universalizability tests use a variety of different satisfaction criteria. For example, consequentialists typically use criteria like "produces at least as much good as any alternative would" or "has at least as much expected value as any alternative." These tend to be aggregative, allowing the addition of value across different agents. Deontologists tend to use non-aggregative criteria like "is not impossible" (Kant's contradiction in conception test), "would make the satisfaction of your ends impossible" (Kant's contradiction in will test), "would disrespect humanity in yourself or another" (Kant's formula of humanity), or "would be reasonable to reject" (Scanlon's contractualist test).
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If one considers two such random matrices which agree on the average value of any quadratic polynomial in the diagonal entries and on the average value of any quartic polynomial in the off-diagonal entries, then Tao and Vu show that the expected value of a large number of functions of the eigenvalues will also coincide, up to an error which is uniformly controllable by the size of the matrix and which becomes arbitrarily small as the size of the matrix increases. Similar results were obtained around the same time by László Erdös, Horng-Tzer Yau, and Jun Yin.Erdős, László; Yau, Horng-Tzer; Yin, Jun.
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Workers are discouraged from switching to jobs where they are more efficient producers, and this immobility of labor resources leads to a lower level of overall productivity and national income. The second implication is that the high risk consumers are more likely to face job lock for fear of losing coverage for their routine medical expenditures (they know their expected value of health bills). Employers offer health insurance benefits to ensure that their workers are healthy and, therefore, productive workers. However, since job lock is common in the high risk employees, employers are ultimately keeping the high risk employees as a part of their company.
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Jacob Bernoulli's Ars Conjectandi (posthumous, 1713) and Abraham De Moivre's The Doctrine of Chances (1718) put probability on a sound mathematical footing, showing how to calculate a wide range of complex probabilities. Bernoulli proved a version of the fundamental law of large numbers, which states that in a large number of trials, the average of the outcomes is likely to be very close to the expected value - for example, in 1000 throws of a fair coin, it is likely that there are close to 500 heads (and the larger the number of throws, the closer to half-and-half the proportion is likely to be).
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For example, suppose there are only two qualities, good cars ("peaches") and bad cars ("lemons"), worth $5000 and $1000, respectively. The buyer knows that half the cars are peaches and half are lemons. If she offers $5000, the sellers of either type surely accept, but the expected value of the car will only be equal to $3000 ($1000 with 50% probability and $5000 with 50% probability), so she will make expected losses of $2000. If she offers $3000, sellers of bad cars will accept but sellers of good cars will not (assuming that sellers are never willing to accept losses to liquidate the value of their cars).
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Econometric theory uses statistical theory and mathematical statistics to evaluate and develop econometric methods. Econometricians try to find estimators that have desirable statistical properties including unbiasedness, efficiency, and consistency. An estimator is unbiased if its expected value is the true value of the parameter; it is consistent if it converges to the true value as the sample size gets larger, and it is efficient if the estimator has lower standard error than other unbiased estimators for a given sample size. Ordinary least squares (OLS) is often used for estimation since it provides the BLUE or "best linear unbiased estimator" (where "best" means most efficient, unbiased estimator) given the Gauss-Markov assumptions.
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The betting round ends when all players have either called the last bet or folded. If all but one player folds on any round, the remaining player collects the pot without being required to reveal their hand. If more than one player remains in contention after the final betting round, a showdown takes place where the hands are revealed, and the player with the winning hand takes the pot. With the exception of initial forced bets, money is only placed into the pot voluntarily by a player who either believes the bet has positive expected value or who is trying to bluff other players for various strategic reasons.
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The Probabilistic Data Association Filter (PDAF) is a statistical approach to the problem of plot association (target-measurement assignment) in a target tracking algorithm. Rather than choosing the most likely assignment of measurements to a target (or declaring the target not detected or a measurement to be a false alarm), the PDAF takes an expected value, which is the minimum mean square error (MMSE) estimate. The PDAF on its own does not confirm nor terminate tracks. Whereas the PDAF is only designed to track a single target in the presence of false alarms and missed detections, the Joint Probabilistic Data Association Filter (JPDAF) can handle multiple targets.
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The expected value is also sometime denoted \langle u\rangle, but it is also seen often with the over-bar notation. Direct Numerical Simulation, or resolving the Navier-Stokes equations completely in (x,y,z,t), is only possible on small computational grids and small time steps when Reynolds numbers are low. Due to computational constraints, simplifications of the Navier-Stokes equations are useful to parameterize turbulence that are smaller than the computational grid, allowing larger computational domains. Reynolds decomposition allows the simplification the Navier–Stokes equations by substituting in the sum of the steady component and perturbations to the velocity profile and taking the mean value.
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Concentration may be played solo either as a leisurely exercise, or with the following scoring method: play as normal, but keep track of the number of non-matching pairs turned over (this may be done using poker chips, pennies or by making marks on a sheet of paper). The object is to clear the tableau in the fewest turns, or to get the lowest possible score. With perfect memorization and using an optimal strategy, the expected number of moves needed for a game with n cards converges to \approx 0.8n , with n \to \infty. For a standard deck of 52 cards, the expected value is \approx 41.4 moves.
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The winning contractor will provide an initial 1,076 systems to supply AH-64 Apache, UH-60 Black Hawk, CH-47 Chinook and future armed scout helicopters. Currently, the DoD plans to award two or more 21-month technology development contracts first, followed by a two-year engineering and manufacturing development phase, with production to begin in 2015 and deployment in 2017. The program has an expected value of $1.5 billion. Competition is fierce for the CIRCM program, with four established industry teams vying for what seems to be one of the few new starts the armed services will pursue in a "bleak" budgetary environment.
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A cubic polynomial regression fit to a simulated data set. The confidence band is a 95% simultaneous confidence band constructed using the Scheffé approach. The goal of regression analysis is to model the expected value of a dependent variable y in terms of the value of an independent variable (or vector of independent variables) x. In simple linear regression, the model : y = \beta_0 + \beta_1 x + \varepsilon, \, is used, where ε is an unobserved random error with mean zero conditioned on a scalar variable x. In this model, for each unit increase in the value of x, the conditional expectation of y increases by β1 units.
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Let M be the maximal value of a Gaussian random function X on the (two-dimensional) sphere. Assume that the expected value of X is 0 (at every point of the sphere), and the standard deviation of X is 1 (at every point of the sphere). Then, for large a>0, P(M>a) is close to C a \exp(-a^2/2) + 2P(\xi>a), where \xi is distributed N(0,1) (the standard normal distribution), and C is a constant; it does not depend on a, but depends on the correlation function of X (see below). The relative error of the approximation decays exponentially for large a.
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In most applications, the entire budget would be used up, because any unspent funds would represent unobtained potential utility. In these situations, the intertemporal budget constraint is effectively an equality constraint. In an intertemporal consumption model, the sum of utilities from expenditures made at various times in the future, these utilities discounted back to the present at the consumer's rate of time preference, would be maximized with respect to the amounts xt consumed in each period, subject to an intertemporal budget constraint. In a model of intertemporal portfolio choice, the objective would be to maximize the expected value or expected utility of final period wealth.
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The most recent major development in backgammon was the addition of the doubling cube. It was first introduced in the 1920s in New York City among members of gaming clubs in the Lower East Side. The cube required players not only to select the best move in a given position, but also to estimate the probability of winning from that position, transforming backgammon into the expected value-driven game played in the 20th and 21st centuries. The popularity of backgammon surged in the mid-1960s, in part due to the charisma of Prince Alexis Obolensky who became known as "The Father of Modern Backgammon".
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Less widely found is best-case performance, but it does have uses: for example, where the best cases of individual tasks are known, they can be used to improve the accuracy of an overall worst-case analysis. Computer scientists use probabilistic analysis techniques, especially expected value, to determine expected running times. The terms are used in other contexts; for example the worst- and best- case outcome of a planned-for epidemic, worst-case temperature to which an electronic circuit element is exposed, etc. Where components of specified tolerance are used, devices must be designed to work properly with the worst- case combination of tolerances and external conditions.
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In probability theory, Wald's equation, Wald's identity or Wald's lemma is an important identity that simplifies the calculation of the expected value of the sum of a random number of random quantities. In its simplest form, it relates the expectation of a sum of randomly many finite-mean, independent and identically distributed random variables to the expected number of terms in the sum and the random variables' common expectation under the condition that the number of terms in the sum is independent of the summands. The equation is named after the mathematician Abraham Wald. An identity for the second moment is given by the Blackwell–Girshick equation.
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Nicolas Bernoulli described the St. Petersburg paradox (involving infinite expected values) in 1713, prompting two Swiss mathematicians to develop expected utility theory as a solution. The theory can also more accurately describe more realistic scenarios (where expected values are finite) than expected value alone. In 1728, Gabriel Cramer, in a letter to Nicolas Bernoulli, wrote, "the mathematicians estimate money in proportion to its quantity, and men of good sense in proportion to the usage that they may make of it." In 1738, Nicolas' cousin Daniel Bernoulli, published the canonical 18th Century description of this solution in Specimen theoriae novae de mensura sortis or Exposition of a New Theory on the Measurement of Risk.
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Despite its theoretical importance, critics of MPT question whether it is an ideal investment tool, because its model of financial markets does not match the real world in many ways. The risk, return, and correlation measures used by MPT are based on expected values, which means that they are statistical statements about the future (the expected value of returns is explicit in the above equations, and implicit in the definitions of variance and covariance). Such measures often cannot capture the true statistical features of the risk and return which often follow highly skewed distributions (e.g. the log-normal distribution) and can give rise to, besides reduced volatility, also inflated growth of return.
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Litigation risk analysis is a growing practice by lawyers, mediators, and other alternative dispute resolution (ADR) professionals. When applied in mediation settings, litigation risk analysis is used to determine litigated best alternative to negotiated agreement (BATNA) and worst alternative to negotiated agreement (WATNA) scenarios based upon the probabilities and possible outcomes of continuing to litigate the case rather than settle. The process of performing a litigation risk analysis by mediators has been hampered by the need for mediators to physically draw out the decision tree and perform calculations to arrive at an expected value (EV). However, there have been calls for more mediators to adopt the practice of performing such an analysis.
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It is often the case that a person, faced with real-world gambles with money, does not act to maximize the expected value of their dollar assets. For example, a person who only possesses $1000 in savings may be reluctant to risk it all for a 20% chance odds to win $10,000, even though :20\%(\$10\,000)+80\%(\$0) = \$2000 > 100\%(\$1000) However, if the person is VNM-rational, such facts are automatically accounted for in their utility function u. In this example, we could conclude that :20\%u(\$10\,000)+80\%u(\$0) < u(\$1000) where the dollar amounts here really represent outcomes (cf. "value"), the three possible situations the individual could face.
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During his time there he produced instructional poker videos where he shared his knowledge of the game. His mathematical approach to poker, as well as his pursuit of constant improvement and self-analysis are what makes Cole South stand out as a poker player: Poker is a game based on mathematics (probability and expectation in particular) and should be treated this way. Bets, folds, raises, bluffs, and all poker decisions should be evaluated solely on their expected value, not because a certain action "feels right" or "gives you information". As a mathematician this is perfectly clear to me, but it isn't evident to casual players and this is what makes poker profitable.
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While "shooting", a portion of powder flows into exhaust without being deposited onto the electrostatic precipitators (ESP), and during collection of powder which is done by brushing it off, powder loss occurs which causes deviation of mass of collected powder from theoretically expected value. In laboratory settings, production rates can range from 10 to 300 g/hour, producing uniform, unaggregated nanoparticles with APS between 15 and 100 nm. Commercially, Nanocerox holds an exclusive license for LF-FSP and can produce 4 kg/hour quantities via the continuous process. Typically, the solvent serves as the fuel; thus cost and solubility issues leads to use of ethanol or other "low cost" alcohols to dissolve the precursors.
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The previous analysis calculates expected value, but we can ask another question: what is the chance that one can play a casino game using the martingale strategy, and avoid the losing streak long enough to double one's bankroll. As before, this depends on the likelihood of losing 6 roulette spins in a row assuming we are betting red/black or even/odd. Many gamblers believe that the chances of losing 6 in a row are remote, and that with a patient adherence to the strategy they will slowly increase their bankroll. In reality, the odds of a streak of 6 losses in a row are much higher than many people intuitively believe.
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To draft this model, the company must possess knowledge of three parameters: # how much the data is worth; # how much the data is at risk; # the probability an attack on the data is going to be successful. This last parameter, Gordon and Loeb defined as vulnerability. These three parameters are multipled together to provide the median money loss with no security investment. Ideal level of investment in company computer security, given decreasing incremental returns From the model we can gather that the amount of money a company spends in protecting information should, in most cases, be only a small fraction of the predicted loss (for example, expected value of a loss following a security breach).
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Mankind's knowledge of star formations will naturally lead one to select the stars of same age and size, and so on, to resolve this problem. In other cases, one's lack of knowledge of the underlying random process then makes the use of Bayesian reasoning less useful. Less accurate, if the knowledge of the possibilities is very unstructured, thereby necessarily having more nearly uniform prior probabilities (by the principle of indifference). Less certain too, if there are effectively few subjective prior observations, and thereby a more nearly minimal total of pseudocounts, giving fewer effective observations, and so a greater estimated variance in expected value, and probably a less accurate estimate of that value.
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In statistics, Gibbs sampling or a Gibbs sampler is a Markov chain Monte Carlo (MCMC) algorithm for obtaining a sequence of observations which are approximated from a specified multivariate probability distribution, when direct sampling is difficult. This sequence can be used to approximate the joint distribution (e.g., to generate a histogram of the distribution); to approximate the marginal distribution of one of the variables, or some subset of the variables (for example, the unknown parameters or latent variables); or to compute an integral (such as the expected value of one of the variables). Typically, some of the variables correspond to observations whose values are known, and hence do not need to be sampled.
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Time lags are far more common on finish-to- start and finish-to-finish relationships ("Wait for cement to dry") than on SS relationships. Critical path drag is often combined with an estimate of the increased cost and/or reduced expected value of the project due to each unit of the critical path's duration. This allows such cost to be attributed to individual critical path activities through their respective drag amounts (i.e., the activity's drag cost). If the cost of each unit of time in the diagram above is $10,000, the drag cost of E would be $200,000, B would be $150,000, A would be $100,000, and C and D $50,000 each.
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An alternate, although less common approach, is to apply a "fundamental valuation" method, such as the "T-model", which instead relies on accounting information. (Other methods of discounting, such as hyperbolic discounting, are studied in academia and said to reflect intuitive decision- making, but are not generally used in industry. In this context the above is referred to as "exponential discounting".) Note that the terminology "expected return", although formally the mathematical expected value, is often used interchangeably with the above, where "expected" means "required" or “demanded” in the corresponding sense. The method may also be modified by industry, for example different have been described when choosing a discount rate in a healthcare setting.
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In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between the estimated values and the actual value. MSE is a risk function, corresponding to the expected value of the squared error loss. That MSE is almost always strictly positive can be attributed to either randomness, or the fact that the estimator does not account for information that could produce a more accurate estimate. The MSE is a measure of the quality of an estimator—it is always non-negative, and values closer to zero are better.
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In using a normal probability plot, the quantiles one uses are the rankits, the quantile of the expected value of the order statistic of a standard normal distribution. More generally, Shapiro–Wilk test uses the expected values of the order statistics of the given distribution; the resulting plot and line yields the generalized least squares estimate for location and scale (from the intercept and slope of the fitted line).Testing for Normality, by Henry C. Thode, CRC Press, 2002, , p. 31 Although this is not too important for the normal distribution (the location and scale are estimated by the mean and standard deviation, respectively), it can be useful for many other distributions.
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When performing statistical analysis, independent variables that affect a particular dependent variable are said to be orthogonal if they are uncorrelated, since the covariance forms an inner product. In this case the same results are obtained for the effect of any of the independent variables upon the dependent variable, regardless of whether one models the effects of the variables individually with simple regression or simultaneously with multiple regression. If correlation is present, the factors are not orthogonal and different results are obtained by the two methods. This usage arises from the fact that if centered by subtracting the expected value (the mean), uncorrelated variables are orthogonal in the geometric sense discussed above, both as observed data (i.e.
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Magriel created the "M Principle" (better known since as the M-ratio) - a theory elaborated on at great length in the book Harrington on Hold'em Volume II by former WSOP Champion "Action" Dan Harrington and Bill Robertie. The theory explains at which stages of tournaments expected value exists to make moves on other players, depending on the ratio between chip stack sizes and antes. While playing poker, Magriel often shouted "Quack quack!" while making a bet, usually to declare a bet which had a numerical value beginning in 22 (e.g.: 2200, 22000.) This is a reference to his nickname, X-22, since a pair of 2's are known in backgammon as "double ducks" and poker as ducks.
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The transactional interpretation of quantum mechanics (TIQM) by John G. Cramer is an interpretation of quantum mechanics inspired by the Wheeler–Feynman absorber theory. It describes the collapse of the wave function as resulting from a time-symmetric transaction between a possibility wave from the source to the receiver (the wave function) and a possibility wave from the receiver to source (the complex conjugate of the wave function). This interpretation of quantum mechanics is unique in that it not only views the wave function as a real entity, but the complex conjugate of the wave function, which appears in the Born rule for calculating the expected value for an observable, as also real.
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DecideIT is a decision-making software that is based on multi-criteria decision making (MCDM) and the multi-attribute value theory (MAVT). It supports both the modelling and evaluation of value trees for multi-attribute decision problems as well as decision trees for evaluating decisions under risk. The software implements the Delta MCDM method and is therefore able to handle imprecise statements in terms of intervals, rankings, and comparisons . Earlier versions employed a so-called contraction analysis approach to evaluate decision problems with imprecise information, but as from DecideIT 3, the software supports second-order probabilities which enables a more discriminative power and more informative means for decision evaluation when expected value intervals are overlapping .
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The distinction between ambiguity aversion and risk aversion is important but subtle. Risk aversion comes from a situation where a probability can be assigned to each possible outcome of a situation and it is defined by the preference between a risky alternative and its expected value. Ambiguity aversion applies to a situation when the probabilities of outcomes are unknown (Epstein 1999) and it is defined through the preference between risky and ambiguous alternatives, after controlling for preferences over risk. Using the traditional two-urn Ellsberg choice, urn A contains 50 red balls and 50 blue balls while urn B contains 100 total balls (either red or blue) but the number of each is unknown.
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Hacking, Ian. (1983) "19th-century Cracks in the Concept of Determinism", Journal of the History of Ideas, 44 (3), 455-475 Thereafter, it was known under both names, but the "law of large numbers" is most frequently used. After Bernoulli and Poisson published their efforts, other mathematicians also contributed to refinement of the law, including Chebyshev, Markov, Borel, Cantelli and Kolmogorov and Khinchin. Markov showed that the law can apply to a random variable that does not have a finite variance under some other weaker assumption, and Khinchin showed in 1929 that if the series consists of independent identically distributed random variables, it suffices that the expected value exists for the weak law of large numbers to be true.
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Marcus Hutter's universal artificial intelligence builds upon Solomonoff's mathematical formalization of the razor to calculate the expected value of an action. There are various papers in scholarly journals deriving formal versions of Occam's razor from probability theory, applying it in statistical inference, and using it to come up with criteria for penalizing complexity in statistical inference. PapersChris S. Wallace and David M. Boulton; Computer Journal, Volume 11, Issue 2, 1968 Page(s):185–194, "An information measure for classification."Chris S. Wallace and David L. Dowe; Computer Journal, Volume 42, Issue 4, Sep 1999 Page(s):270–283, "Minimum Message Length and Kolmogorov Complexity." have suggested a connection between Occam's razor and Kolmogorov complexity.
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One possible and common answer to this question is to find a path with the minimum expected travel time. The main advantage of using this approach is that efficient shortest path algorithms introduced for the deterministic networks can be readily employed to identify the path with the minimum expected travel time in a stochastic network. However, the resulting optimal path identified by this approach may not be reliable, because this approach fails to address travel time variability. To tackle this issue some researchers use distribution of travel time instead of expected value of it so they find the probability distribution of total travelling time using different optimization methods such as dynamic programming and Dijkstra's algorithm .
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The generalized functional linear model (GFLM) is an extension of the generalized linear model (GLM) that allows one to regress univariate responses of various types (continuous or discrete) on functional predictors, which are mostly random trajectories generated by a square-integrable stochastic processes. Similarly to GLM, a link function relates the expected value of the response variable to a linear predictor, which in case of GFLM is obtained by forming the scalar product of the random predictor function X with a smooth parameter function \beta . Functional Linear Regression, Functional Poisson Regression and Functional Binomial Regression, with the important Functional Logistic Regression included, are special cases of GFLM. Applications of GFLM include classification and discrimination of stochastic processes and functional data.
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Thus, it represents the average amount one expects to win per bet if bets with identical odds are repeated many times. A game or situation in which the expected value for the player is zero (no net gain nor loss) is called a fair game. The attribute fair refers not to the technical process of the game, but to the chance balance house (bank)–player. Even though the randomness inherent in games of chance would seem to ensure their fairness (at least with respect to the players around a table—shuffling a deck or spinning a wheel do not favor any player except if they are fraudulent), gamblers always search and wait for irregularities in this randomness that will allow them to win.
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Although the sampled values represent the joint distribution over all variables, the nuisance variables can simply be ignored when computing expected values or modes; this is equivalent to marginalizing over the nuisance variables. When a value for multiple variables is desired, the expected value is simply computed over each variable separately. (When computing the mode, however, all variables must be considered together.) Supervised learning, unsupervised learning and semi- supervised learning (aka learning with missing values) can all be handled by simply fixing the values of all variables whose values are known, and sampling from the remainder. For observed data, there will be one variable for each observation—rather than, for example, one variable corresponding to the sample mean or sample variance of a set of observations.
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In 1998, the USGS estimated that the 1002 area of the Arctic National Wildlife Refuge contains a total of between 5.7 and of undiscovered, technically recoverable oil, with a mean estimate of , of which falls within the Federal portion of the ANWR 1002 Area. In May 2008 the EIA used this assessment to estimate the potential cumulative production of the 1002 area of ANWR to be a maximum of from 2018 to 2030. This estimate is a best case scenario of technically recoverable oil during the area's primary production years if legislation were passed in 2008 to allow drilling. A 2002 assessment concluded that the National Petroleum Reserve–Alaska contains between 6.7 and of oil, with a mean (expected) value of .
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The data sets in the Anscombe's quartet are designed to have approximately the same linear regression line (as well as nearly identical means, standard deviations, and correlations) but are graphically very different. This illustrates the pitfalls of relying solely on a fitted model to understand the relationship between variables. A fitted linear regression model can be used to identify the relationship between a single predictor variable xj and the response variable y when all the other predictor variables in the model are "held fixed". Specifically, the interpretation of βj is the expected change in y for a one-unit change in xj when the other covariates are held fixed—that is, the expected value of the partial derivative of y with respect to xj.
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In probability theory and statistics, a central moment is a moment of a probability distribution of a random variable about the random variable's mean; that is, it is the expected value of a specified integer power of the deviation of the random variable from the mean. The various moments form one set of values by which the properties of a probability distribution can be usefully characterized. Central moments are used in preference to ordinary moments, computed in terms of deviations from the mean instead of from zero, because the higher-order central moments relate only to the spread and shape of the distribution, rather than also to its location. Sets of central moments can be defined for both univariate and multivariate distributions.
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Investors typically do not like to hold stocks during a financial crisis, and thus investors may sell stocks until the price drops enough so that the expected return compensates for this risk. How efficient markets are (and are not) linked to the random walk theory can be described through the fundamental theorem of asset pricing. This theorem states that, in the absence of arbitrage, the price of any stock is given by :P_t = E_t[M_{t+1} (P_{t+1}+D_{t+1})] where E_{t} is the expected value given information at time t, M_{t+1} is the stochastic discount factor, and D_{t+1}is the dividend the stock pays next period. Note that this equation does not generally imply a random walk.
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Rather than a coincidence that the two apparently contradictory approaches to economic modeling were developed at GSIA at the same time, it is more likely that fruitful interaction in the quest to answer a common set of problems led the two researchers to two different solutions. In an earlier work, Herb Simon had shown that with quadratic costs and under a certain set of assumptions about the probability distributions, optimal decision rules for production and inventories would be linear functions of the variables describing the state. In his model, firms only needed to take into account the expected value and ignore all higher moments of the probability distribution of future sales. This result, known as certainty equivalence, drastically reduces the computational burden on a representative decision maker.
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Would it exist, the CVSO 30 b would be a hot Jupiter planet orbiting the T Tauri star, with 6.2 Jupiter masses. Direct imaging of suspected CVSO 30 c, with calculated mass equal to 4.7 Jupiter's, has been achieved through photometric and spectroscopic high contrast observations carried out with the Very Large Telescope located in Chile, the Keck Observatory in Hawaii and the Calar Alto Observatory in Spain. However, the colors of the object suggest that it may actually be a background star, such as a K-type giant or a M-type subdwarf. By 2020, phase of "dips" caused by suspected planet CVSO 30 b, have drifted nearly 180 degrees from expected value, thus ruling out the existence of planet.
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CBO-Public Debt Under "Extended" and "Alternate" Scenarios A credit rating is issued by a credit rating agency (CRA). A credit rating assigned to U.S. sovereign debt is an expression of how likely the assigning CRA thinks it is that the U.S. will pay back its debts. A credit rating assigned to U.S. sovereign debt also influences the interest rates the U.S. will have to pay on its debt; if its debtholders know the debt will be paid back, they do not have to price the chance of default into the interest rate. However, these ratings sometimes measure different things; for instance Moody's considers the expected value of the debt in the event of a default in addition to the probability of default.
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An extremely well-studied formulation in stochastic control is that of linear quadratic Gaussian control. Here the model is linear, the objective function is the expected value of a quadratic form, and the disturbances are purely additive. A basic result for discrete-time centralized systems with only additive uncertainty is the certainty equivalence property: that the optimal control solution in this case is the same as would be obtained in the absence of the additive disturbances. This property is applicable to all centralized systems with linear equations of evolution, quadratic cost function, and noise entering the model only additively; the quadratic assumption allows for the optimal control laws, which follow the certainty-equivalence property, to be linear functions of the observations of the controllers.
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The removed nodes are then reinserted into the network with their original weights. In the training stages, the probability that a hidden node will be dropped is usually 0.5; for input nodes, however, this probability is typically much lower, since information is directly lost when input nodes are ignored or dropped. At testing time after training has finished, we would ideally like to find a sample average of all possible 2^n dropped-out networks; unfortunately this is unfeasible for large values of n. However, we can find an approximation by using the full network with each node's output weighted by a factor of p, so the expected value of the output of any node is the same as in the training stages.
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Fano noise is a fluctuation of an electric charge obtained in a detector (in spite of constant value of the measured quantity, which is usually an energy), arising from processes in the detector. It was first described by Ugo Fano in 1947, as a fluctuation of amount of ion pairs produced by a charged particle of high energy in a gas. The amount of the ion pairs is proportional to the energy the particle loses in the gas, but with some error - due to the Fano noise. Surprisingly, the noise is usually smaller than a Poisson distribution noise (in which the variance is equal to the value - note the variance is average squared distance from the expected value), showing there is an interaction between ionization acts.
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Information content is defined as the logarithm of the reciprocal of the probability that a system is in a specific microstate, and the information entropy of a system is the expected value of the system's information content. This definition of entropy is equivalent to the standard Gibbs entropy used in classical physics. Applying this definition to a physical system leads to the conclusion that, for a given energy in a given volume, there is an upper limit to the density of information (the Bekenstein bound) about the whereabouts of all the particles which compose matter in that volume. In particular, a given volume has an upper limit of information it can contain, at which it will collapse into a black hole.
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A key feature of minimax decision making is being non-probabilistic: in contrast to decisions using expected value or expected utility, it makes no assumptions about the probabilities of various outcomes, just scenario analysis of what the possible outcomes are. It is thus robust to changes in the assumptions, as these other decision techniques are not. Various extensions of this non-probabilistic approach exist, notably minimax regret and Info-gap decision theory. Further, minimax only requires ordinal measurement (that outcomes be compared and ranked), not interval measurements (that outcomes include "how much better or worse"), and returns ordinal data, using only the modeled outcomes: the conclusion of a minimax analysis is: "this strategy is minimax, as the worst case is (outcome), which is less bad than any other strategy".
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An early book on compound interest , Institute and Faculty of Actuaries further developed by Johan de Witt and Edmond Halley.) An immediate extension is to combine probabilities with present value, leading to the expected value criterion which sets asset value as a function of the sizes of the expected payouts and the probabilities of their occurrence, X_{s} and p_{s} respectively. (These ideas originate with Blaise Pascal and Pierre de Fermat in 1654.) This decision method, however, fails to consider risk aversion ("as any student of finance knows"). In other words, since individuals receive greater utility from an extra dollar when they are poor and less utility when comparatively rich, the approach is to therefore "adjust" the weight assigned to the various outcomes ("states") correspondingly, Y_{s}. See Indifference price.
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Ronald Fisher in 1913 Early users of maximum likelihood were Carl Friedrich Gauss, Pierre-Simon Laplace, Thorvald N. Thiele, and Francis Ysidro Edgeworth. However, its widespread use rose between 1912 and 1922 when Ronald Fisher recommended, widely popularized, and carefully analyzed maximum- likelihood estimation (with fruitless attempts at proofs). Maximum-likelihood estimation finally transcended heuristic justification in a proof published by Samuel S. Wilks in 1938, now called Wilks' theorem. The theorem shows that the error in the logarithm of likelihood values for estimates from multiple independent observations is asymptotically χ 2-distributed, which enables convenient determination of a confidence region around any estimate of the parameters. The only difficult part of Wilks’ proof depends on the expected value of the Fisher information matrix, which is provided by a theorem proven by Fisher.
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As the housing market began to collapse and defaults on subprime mortgages began to mount, disputes between Hubler's group and their counterparties began to emerge over the value of the bonds and CDOs that had been subject to credit default swaps. When notified by the counterparties that the value of the CDOs had dropped to levels warranting a payout, Hubler disagreed on the assessment stating that the GPCG's models indicated that the CDOs were worth most of their expected value. Had Hubler conceded the drop in value of the CDOs earlier, the GPCG's losses may have been contained to a relatively small fraction of their overall risk. However, because of his reluctance to follow the procedures outlined in the credit default swaps, GPCG and Morgan Stanley's position worsened over the coming months.
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The distances calculated by this method must be linear; the linearity criterion for distances requires that the expected values of the branch lengths for two individual branches must equal the expected value of the sum of the two branch distances – a property that applies to biological sequences only when they have been corrected for the possibility of back mutations at individual sites. This correction is done through the use of a substitution matrix such as that derived from the Jukes–Cantor model of DNA evolution. The least-squares criterion applied to these distances is more accurate but less efficient than the neighbor-joining methods. An additional improvement that corrects for correlations between distances that arise from many closely related sequences in the data set can also be applied at increased computational cost.
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Both systems have the same state dimension. A deeper statement of the separation principle is that the LQG controller is still optimal in a wider class of possibly nonlinear controllers. That is, utilizing a nonlinear control scheme will not improve the expected value of the cost functional. This version of the separation principle is a special case of the separation principle of stochastic control which states that even when the process and output noise sources are possibly non-Gaussian martingales, as long as the system dynamics are linear, the optimal control separates into an optimal state estimator (which may no longer be a Kalman filter) and an LQR regulator.. In the classical LQG setting, implementation of the LQG controller may be problematic when the dimension of the system state is large.
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The Fitch–Margoliash method uses a weighted least squares method for clustering based on genetic distance. Closely related sequences are given more weight in the tree construction process to correct for the increased inaccuracy in measuring distances between distantly related sequences. The distances used as input to the algorithm must be normalized to prevent large artifacts in computing relationships between closely related and distantly related groups. The distances calculated by this method must be linear; the linearity criterion for distances requires that the expected values of the branch lengths for two individual branches must equal the expected value of the sum of the two branch distances - a property that applies to biological sequences only when they have been corrected for the possibility of back mutations at individual sites.
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This is mathematically convenient, as the standard exponential distribution has both the expected value and the standard deviation equal to 2Ne. Therefore, although the expected time to coalescence is 2Ne, actual coalescence times have a wide range of variation. Note that coalescent time is the number of preceding generations where the coalescence took place and not calendar time, though an estimation of the latter can be made multiplying 2Ne with the average time between generations. The above calculations apply equally to a diploid population of effective size Ne (in other words, for a non-recombining segment of DNA, each chromosome can be treated as equivalent to an independent haploid individual; in the absence of inbreeding, sister chromosomes in a single individual are no more closely related than two chromosomes randomly sampled from the population).
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In mathematical finance, a risk-neutral measure (also called an equilibrium measure, or equivalent martingale measure) is a probability measure such that each share price is exactly equal to the discounted expectation of the share price under this measure. This is heavily used in the pricing of financial derivatives due to the fundamental theorem of asset pricing, which implies that in a complete market a derivative's price is the discounted expected value of the future payoff under the unique risk-neutral measure. Such a measure exists if and only if the market is arbitrage-free. The easiest way to remember what the risk-neutral measure is, or to explain it to a probability generalist who might not know much about finance, is to realize that it is: # The probability measure of a transformed random variable.
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A value breakdown structure (VBS) is a project management technique introduced by Stephen Devaux as part of the total project control (TPC) approach to project and program value analysis. A work breakdown structure (WBS) in project management and systems engineering is a deliverable-oriented decomposition of a project into smaller components into a tree structure that represents how the work of the project will create the components of the final product. Resources and cost are typically inserted into the activities in a WBS, and summed to create a budget both for summary levels (often called "work packages") and for the whole project or program. Similarly, the expected value-added of each activity and/or component of the project (or projects within a program) are inserted into the VBS.
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Types of direct current The term DC is used to refer to power systems that use only one polarity of voltage or current, and to refer to the constant, zero-frequency, or slowly varying local mean value of a voltage or current. For example, the voltage across a DC voltage source is constant as is the current through a DC current source. The DC solution of an electric circuit is the solution where all voltages and currents are constant. It can be shown that any stationary voltage or current waveform can be decomposed into a sum of a DC component and a zero-mean time-varying component; the DC component is defined to be the expected value, or the average value of the voltage or current over all time.
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A scaled version of the curve is the probability density function of the Cauchy distribution. This is the probability distribution on the random variable x determined by the following random experiment: for a fixed point p above the x-axis, choose uniformly at random a line through p, and let x be the coordinate of the point where this random line crosses the axis. The Cauchy distribution has a peaked distribution visually resembling the normal distribution, but its heavy tails prevent it from having an expected value by the usual definitions, despite its symmetry. In terms of the witch itself, this means that the x-coordinate of the centroid of the region between the curve and its asymptotic line is not well-defined, despite this region's symmetry and finite area.
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In 2011, the ARCADE 2 researchers reported, "Correcting for instrumental systematic errors in measurements such as ARCADE 2 is always a primary concern. We emphasize that we detect residual emission at 3 GHz with the ARCADE 2 data, but the result is also independently detected by a combination of low-frequency data and FIRAS." The ARCADE 2 science team came to the following conclusion concerning the unexpected residual emission at 3 GHz: Radio waves have frequencies from 30 Hz to 300 GHz. The term space roar has been used to indicate the hypothesis that the ARCADE 2 results indicate that the actual faint end of the emission distribution of known sources is significantly different from the expected value predicted by the Lambda-CDM model given the known sources of emission.
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Car collecting as an investment can be rewarding but most serious investment collectors seek rare or exotic cars and original unmodified cars hold a more stable price. Collecting as an investment requires expertise beyond enthusiast collecting and the standard of quality is far higher as well as a need for investment protection such as storage and maintenance. A short-term investment collector must be able to find a vehicle that has market value that is expected to rise in the foreseeable near future. A long-term investment collector would be less interested in any short-term value seeking to capitalize on an expected value rise over a period of years and a vehicle must have certain intrinsic values that are common to other investors or collectors of both short and long term.
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Mean square quantization error (MSQE) is a figure of merit for the process of analog to digital conversion. In this conversion process, analog signals in a continuous range of values are converted to a discrete set of values by comparing them with a sequence of thresholds. The quantization error of a signal is the difference between the original continuous value and its discretization, and the mean square quantization error (given some probability distribution on the input values) is the expected value of the square of the quantization errors. Mathematically, suppose that the lower threshold for inputs that generate the quantized value q_i is t_{i-1}, that the upper threshold is t_i, that there are k levels of quantization, and that the probability density function for the input analog values is p(x).
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Now, assume type B project returns are also uniformly distributed, but their range is from $50 to $150. Type B project returns also have an expected value of $100, but are more risky. Now assume that the bank knows that two types exist, and even knows what fraction of the potential borrowers applying for loans belong to each group, but cannot tell whether an individual applicant is type A or B. The implication to the bank of the difference in the riskiness of these projects is that each borrower has a different probability of repaying the loan, and this affects the bank's expected return. The bank would thus like to be able to identify (screen) the borrower types, and in the absence of other instruments to do so, it will use the interest rate.
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The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes in a fair way between two players, who have to end their game before it is properly finished. This problem had been debated for centuries, and many conflicting proposals and solutions had been suggested over the years, when it was posed to Blaise Pascal by French writer and amateur mathematician Chevalier de Méré in 1654. Méré claimed that this problem couldn't be solved, and that it showed just how flawed mathematics was when it came to its application to the real world. Pascal, being a mathematician, was provoked and determined to solve the problem once and for all.
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The joint probabilistic data-association filter (JPDAF) is a statistical approach to the problem of plot association (target-measurement assignment) in a target tracking algorithm. Like the probabilistic data association filter (PDAF), rather than choosing the most likely assignment of measurements to a target (or declaring the target not detected or a measurement to be a false alarm), the PDAF takes an expected value, which is the minimum mean square error (MMSE) estimate for the state of each target. At each time, it maintains its estimate of the target state as the mean and covariance matrix of a multivariate normal distribution. However, unlike the PDAF, which is only meant for tracking a single target in the presence of false alarms and missed detections, the JPDAF can handle multiple target tracking scenarios.
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Improved interoperability with major allies allowed the Canadian Forces to gain insight on leading edge practices in composites, manufacturing and logistics, and offered the ability to recoup some investment if the Government of Canada did decide to purchase the F-35. As a result of the Government of Canada's investment in the JSF project, 144 contracts were awarded to Canadian companies, universities, and government facilities. Financially, the contracts are valued at US$490 million for the period 2002 to 2012, with an expected value of US$1.1 billion from current contracts in the period between 2013 and 2023, and a total potential estimated value of Canada's involvement in the JSF project from US$4.8 billion to US$6.8 billion. By 2013 the potential benefits to Canadian firms had risen to $9.9 billion.
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In statistics, the Rao–Blackwell theorem, sometimes referred to as the Rao–Blackwell–Kolmogorov theorem, is a result which characterizes the transformation of an arbitrarily crude estimator into an estimator that is optimal by the mean-squared-error criterion or any of a variety of similar criteria. The Rao–Blackwell theorem states that if g(X) is any kind of estimator of a parameter θ, then the conditional expectation of g(X) given T(X), where T is a sufficient statistic, is typically a better estimator of θ, and is never worse. Sometimes one can very easily construct a very crude estimator g(X), and then evaluate that conditional expected value to get an estimator that is in various senses optimal. The theorem is named after Calyampudi Radhakrishna Rao and David Blackwell.
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In economics, game theory, and decision theory, the expected utility hypothesis—concerning people's preferences with regard to choices that have uncertain outcomes (probabilistic)—states that the subjective value associated with an individual's gamble is the statistical expectation of that individual's valuations of the outcomes of that gamble, where these valuations may differ from the dollar value of those outcomes. The introduction of St. Petersburg Paradox by Daniel Bernoulli in 1738 is considered the beginnings of the hypothesis. This hypothesis has proven useful to explain some popular choices that seem to contradict the expected value criterion (which takes into account only the sizes of the payouts and the probabilities of occurrence), such as occur in the contexts of gambling and insurance. The von Neumann–Morgenstern utility theorem provides necessary and sufficient conditions under which the expected utility hypothesis holds.
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In Bayesian credibility, we separate each class (B) and assign them a probability (Probability of B). Then we find how likely our experience (A) is within each class (Probability of A given B). Next, we find how likely our experience was over all classes (Probability of A). Finally, we can find the probability of our class given our experience. So going back to each class, we weight each statistic with the probability of the particular class given the experience. Bühlmann credibility works by looking at the Variance across the population. More specifically, it looks to see how much of the Total Variance is attributed to the Variance of the Expect Values of each class (Variance of the Hypothetical Mean), and how much is attributed to the Expected Variance over all classes (Expected Value of the Process Variance).
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Once n-1 F-light edges have been added to H none of the subsequent edges considered are F-light by the cycle property. Thus, the number of F-light edges in G is bounded by the number of F-light edges considered for H before n-1 F-light edges are actually added to H. Since any F-light edge is added with probability p this is equivalent to flipping a coin with probability p of coming up heads until n-1 heads have appeared. The total number of coin flips is equal to the number of F-light edges in G. The distribution of the number of coin flips is given by the inverse binomial distribution with parameters n-1 and p. For these parameters the expected value of this distribution is (n-1)/p.
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This can be visualized by imagining that the observations in the sample are evenly spaced throughout the range, with additional observations just outside the range at 0 and N + 1\. If starting with an initial gap between 0 and the lowest observation in the sample (the sample minimum), the average gap between consecutive observations in the sample is (m - k)/k; the -k being because the observations themselves are not counted in computing the gap between observations.. A derivation of the expected value and the variance of the sample maximum are shown in the page of the discrete uniform distribution. This philosophy is formalized and generalized in the method of maximum spacing estimation; a similar heuristic is used for plotting position in a Q–Q plot, plotting sample points at , which is evenly on the uniform distribution, with a gap at the end.
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An informative prior expresses specific, definite information about a variable. An example is a prior distribution for the temperature at noon tomorrow. A reasonable approach is to make the prior a normal distribution with expected value equal to today's noontime temperature, with variance equal to the day- to-day variance of atmospheric temperature, or a distribution of the temperature for that day of the year. This example has a property in common with many priors, namely, that the posterior from one problem (today's temperature) becomes the prior for another problem (tomorrow's temperature); pre-existing evidence which has already been taken into account is part of the prior and, as more evidence accumulates, the posterior is determined largely by the evidence rather than any original assumption, provided that the original assumption admitted the possibility of what the evidence is suggesting.
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The main purpose of the VBS is to prioritize components and work by the value they are expected to add, and to ensure that the value of the project investment is not reduced by the inclusion of work which has a value-added that is less than its true cost, which is the sum of its resource costs and its drag cost. This can often happen if the project's critical path changes so that different activities suddenly acquire critical path drag and drag cost: an optional activity that adds $10,000 to the expected value of the project and has a budget of $5,000 may make sense when it can be performed off the critical path but should probably be jettisoned if on the critical path if they now have a negative value-added due to a drag cost of more than $5,000.
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For any fixed choice of a value in a given set of numbers, if one randomly permutes the numbers and forms a binary tree from them as described above, the expected value of the length of the path from the root of the tree to is at most , where "" denotes the natural logarithm function and the introduces big O notation. For, the expected number of ancestors of is by linearity of expectation equal to the sum, over all other values in the set, of the probability that is an ancestor of . And a value is an ancestor of exactly when is the first element to be inserted from the elements in the interval . Thus, the values that are adjacent to in the sorted sequence of values have probability of being an ancestor of , the values one step away have probability , etc.
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In some games such as video poker, blackjack, or Caribbean stud poker, it is possible to compute an optimal playing strategy based on the average payoff (the amount of payoff times the chance of payoff). Because the jackpot of a progressive game constantly grows, it sometimes exceeds the break-even point for players, such that the jackpot wager becomes a "positive expectation bet" for the player, with an average return to player (RTP) of greater than 100%. When the progressive jackpot is less than the break-even point, there is a negative expected value (house edge) for all players. In the long run, with optimal strategy, a player can profit by only playing progressive games when their jackpots are above the break-even point, although the "long run" can be quite long, tens of thousands of plays.
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In a framework similar to Stiglitz and Weiss, one can imagine a group of individuals, prospective borrowers, who want to borrow funds in order to finance a project, which yields uncertain returns. Let there be two types of individuals, who are observationally identical, and only differ in the riskiness of their projects. Assume type A individuals are low risk compared to type B, in the sense that the expected return on type B projects is a mean preserving spread of type A projects; they have the same expected return, but higher variance. For example, imagine that type A returns are uniformly distributed (meaning that all possible values have the same probability of occurring) from $75 to $125, so that the value of type A projects is at least $75 and at most $125, and the expected value (mean) is $100.
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" However, a follow-up study by Bell and McDiarmid shows that Arp's hypothesis about the periodicity in red-shifts cannot be discarded easily. The authors argue (as response to Tang and Zhang (2005) from which the preceding excerpt is taken) that :"The Tang and Zhang (2005) analysis could thus have missed, or misidentified, many of the parent galaxies, which could explain why the pairs they found differed little from what would be expected for a random distribution. In spite of this, although it was not pointed out by these authors, their pairs did show a slight excess near the expected value of 200 kpc….In fact, most of the conclusions reached by Tang and Zhang (2005) appear to have resulted because they have assumed that many of the values [that they have used] are much more accurate than they really are.
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In decision theory, economics, and finance, a two-moment decision model is a model that describes or prescribes the process of making decisions in a context in which the decision-maker is faced with random variables whose realizations cannot be known in advance, and in which choices are made based on knowledge of two moments of those random variables. The two moments are almost always the mean—that is, the expected value, which is the first moment about zero—and the variance, which is the second moment about the mean (or the standard deviation, which is the square root of the variance). The most well- known two-moment decision model is that of modern portfolio theory, which gives rise to the decision portion of the Capital Asset Pricing Model; these employ mean-variance analysis, and focus on the mean and variance of a portfolio's final value.
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In economics and other social sciences, preference refers to the set of assumptions related to ordering some alternatives, based on the degree of happiness, satisfaction, gratification, enjoyment, or utility they provide, a process which results in an optimal "choice" (whether real or imagined). Although economists are usually not interested in choices or preferences in themselves, they are interested in the theory of choice because it serves as a background for empirical demand analysis. The so-called Expected Utility Theory (EUT), which was introduced by John von Neumann and Oskar Morgenstern in 1944, explains that so long as an agent's preferences over risky options follow a set of axioms, then he is maximizing the expected value of a utility function. This theory specifically identified four axioms that determine an individual's preference when selecting an alternative out of a series of choices that maximizes expected utility for him.
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The input to this voting system consists of the agents' ordinal preferences over outcomes (not lotteries over outcomes), but a relation on the set of lotteries is constructed in the following way: if p and q are different lotteries over outcomes, p\succ q if the expected value of the margin of victory of an outcome selected with distribution p in a head-to-head vote against an outcome selected with distribution q is positive. While this relation is not necessarily transitive, it does always contain at least one maximal element. It is possible that several such maximal lotteries exist, but unicity can be proven in the case where the margins between any pair of alternatives is always an odd number.Gilbert Laffond, Jean-François Laslier and Michel Le Breton A theorem on two–player symmetric zero–sum games Journal of Economic Theory 72: 426–431, 1997.
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The Hammersley set, a low-discrepancy set of points obtained from the van der Corput sequence Although Roth's work on Diophantine approximation led to the highest recognition for him, it is his research on irregularities of distribution that (according to an obituary by William Chen and Bob Vaughan) he was most proud of. His 1954 paper on this topic laid the foundations for modern discrepancy theory. It concerns the placement of n points in a unit square so that, for every rectangle bounded between the origin and a point of the square, the area of the rectangle is well- approximated by the number of points in it. Roth measured this approximation by the squared difference between the number of points and n times the area, and proved that for a randomly chosen rectangle the expected value of the squared difference is logarithmic in n.
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In statistics and in particular statistical theory, unbiased estimation of a standard deviation is the calculation from a statistical sample of an estimated value of the standard deviation (a measure of statistical dispersion) of a population of values, in such a way that the expected value of the calculation equals the true value. Except in some important situations, outlined later, the task has little relevance to applications of statistics since its need is avoided by standard procedures, such as the use of significance tests and confidence intervals, or by using Bayesian analysis. However, for statistical theory, it provides an exemplar problem in the context of estimation theory which is both simple to state and for which results cannot be obtained in closed form. It also provides an example where imposing the requirement for unbiased estimation might be seen as just adding inconvenience, with no real benefit.
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Prospect Theory (PT) claims that fair gambles (gambles in which the expected value of the current option and all other alternatives are held equal) are unattractive on the gain side but attractive on the loss side. In contrast to EUT, PT is posited as an alternative theory of choice, in which value is assigned to gains and losses rather than to final assets (total wealth), and in which probabilities are replaced by decision weights. In an effort to capture inconsistencies in our preferences, PT offers a non-linear, S-shaped probability-weighted value function, implying that the decision-maker transforms probabilities along a diminishing sensitivity curve, in which the impact of a given change in probability diminishes with its distance from impossibility and certainty. The value function shown is: Predicted utility curve of prospect theory A. Defined on gains and losses rather than on total wealth.
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Consequently, the date for the final release of data was pushed back several times. In the data for the frame-dragging results presented at the April 2007 meeting of the American Physical Society, the random errors were much larger than the theoretical expected value and scattered on both the positive and negative sides of a null result, therefore causing skepticism as to whether any useful data could be extracted in the future to test this effect. In June 2007, a detailed update was released explaining the cause of the problem, and the solution that was being worked on. Although electrostatic patches caused by non-uniform coating of the spheres were anticipated, and were thought to have been controlled for before the experiment, it was subsequently found that the final layer of the coating on the spheres defined two halves of slightly different contact potential, which gave the sphere an electrostatic axis.
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PnL unexplained is a critical metric that regulators and product control within a bank alike pay attention to. PnL attribution is used to test the hypothesis that the risk factors identified for a risky position are sufficient to materially explain the value change expected from the risky position;. Such that if position sensitivities to those risk factors are calculated, then the value change observed over a day can be attributed to the market price change of those risk factors, with the magnitude of the estimated as a sum product of the risk factor sensitivities and the corresponding daily risk factor price change. Any residual P&L; left unexplained (PnL unexplained) would be expected to be small IF the identified risk factors are indeed sufficient to materially explain the expected value change of the position AND if the models used to calculate sensitivities to these risk factors are correct.
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Rounding a number to the nearest integer requires some tie-breaking rule for those cases when is exactly half-way between two integers — that is, when the fraction part of is exactly 0.5. If it were not for the 0.5 fractional parts, the round-off errors introduced by the round to nearest method would be symmetric: for every fraction that gets rounded down (such as 0.268), there is a complementary fraction (namely, 0.732) that gets rounded up by the same amount. When rounding a large set of fixed-point numbers with uniformly distributed fractional parts, the rounding errors by all values, with the omission of those having 0.5 fractional part, would statistically compensate each other. This means that the expected (average) value of the rounded numbers is equal to the expected value of the original numbers when we remove numbers with fractional part 0.5 from the set.
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Faria and Horta also assert that welfare biology could be developed from ecology, with a focus on how the well-being of sentient individuals is affected by their environments. They raise a concern of what they see as the minimization of the importance of animal well-being, caused by widespread speciesist and environmentalist beliefs among life scientists and the general public, which they argue could hamper the development of welfare biology. Faria and Horta conclude that the "expected value of developing welfare biology is extremely high" because of the massive extent of animal suffering in the wild, which refutes commonly conceived idyllic conceptions of nature. Some researchers have emphasised the importance of life history theory to welfare biology, as they argue certain traits of life history may predispose certain individuals to worse welfare outcomes and that this has a strong relationship with habitat fragmentation sensitivity.
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Discrete choice models are motivated using utility theory so as to handle various types of correlated and uncorrelated choices, while binomial regression models are generally described in terms of the generalized linear model, an attempt to generalize various types of linear regression models. As a result, discrete choice models are usually described primarily with a latent variable indicating the "utility" of making a choice, and with randomness introduced through an error variable distributed according to a specific probability distribution. Note that the latent variable itself is not observed, only the actual choice, which is assumed to have been made if the net utility was greater than 0. Binary regression models, however, dispense with both the latent and error variable and assume that the choice itself is a random variable, with a link function that transforms the expected value of the choice variable into a value that is then predicted by the linear predictor.
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It should be noted this is the same tax treatment of direct participation in a qualified pension plan (such as a 401K), again, due to the fact the taxpayer has no tax basis in any of the money in the plan. If the annuity contract is purchased with after-tax dollars, then the contract holder upon annuitization recovers his basis pro-rata in the ratio of basis divided by the expected value, according to the tax regulation Section 1.72-5. (This is commonly referred to as the exclusion ratio.) After the taxpayer has recovered all of his basis, then 100% of the payments thereafter are subject to ordinary income tax. Since the Jobs and Growth Tax Relief Reconciliation Act of 2003, the use of variable annuities as a tax shelter has greatly diminished, because the growth of mutual funds and now most of the dividends of the fund are taxed at long term capital gains rates.
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In the expected utility theory of von Neumann and Morgenstern, four axioms together imply that individuals act in situations of risk as if they maximize the expected value of a utility function. One of the axioms is an independence axiom analogous to the IIA axiom: :If \,L\prec M, then for any \,N and \,p\in(0,1], ::\,pL+(1-p)N \prec pM+(1-p)N, where p is a probability, pL+(1-p)N means a gamble with probability p of yielding L and probability (1-p) of yielding N, and \,L\prec M means that M is preferred over L. This axiom says that if one outcome (or lottery ticket) L is considered to be not as good as another (M), then having a chance with probability p of receiving L rather than N is considered to be not as good as having a chance with probability p of receiving M rather than N.
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Many of the lossless compression techniques used for text also work reasonably well for indexed images, but there are other techniques that do not work for typical text that are useful for some images (particularly simple bitmaps), and other techniques that take advantage of the specific characteristics of images (such as the common phenomenon of contiguous 2-D areas of similar tones, and the fact that color images usually have a preponderance of a limited range of colors out of those representable in the color space). As mentioned previously, lossless sound compression is a somewhat specialized area. Lossless sound compression algorithms can take advantage of the repeating patterns shown by the wave-like nature of the data – essentially using autoregressive models to predict the "next" value and encoding the (hopefully small) difference between the expected value and the actual data. If the difference between the predicted and the actual data (called the error) tends to be small, then certain difference values (like 0, +1, −1 etc.
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Portfolio optimization often takes place in two stages: optimizing weights of asset classes to hold, and optimizing weights of assets within the same asset class. An example of the former would be choosing the proportions placed in equities versus bonds, while an example of the latter would be choosing the proportions of the stock sub-portfolio placed in stocks X, Y, and Z. Equities and bonds have fundamentally different financial characteristics and have different systematic risk and hence can be viewed as separate asset classes; holding some of the portfolio in each class provides some diversification, and holding various specific assets within each class affords further diversification. By using such a two-step procedure one eliminates non- systematic risks both on the individual asset and the asset class level. One approach to portfolio optimization is to specify a von Neumann–Morgenstern utility function defined over final portfolio wealth; the expected value of utility is to be maximized.
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This effect was consistent across age and race cohorts. A similar study among 12,275,033 Swiss found the highest mortality on the actual birthday (17% greater than the expected value), and the effect was largest for those over 80; another study on Swiss data found a 13.8% excess and was able to link this to specific causes: heart attack and stroke (predominant in women) and suicides and accidents (predominant in men), as well as an increase in cancer deaths. Among 25 million Americans who died between 1998 and 2011, 6.7% more people than expected die on their birthday, and the effect was most pronounced at weekends and among the young – among 20 to 29 year olds, the excess was over 25%. An even greater excess was found in the population of Kiev, where between 1990 and 2000 there were 44.4% more deaths than expected among men on their birthdays and 36.2% more than expected among women.
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An individual who lends money for repayment at a later point in time expects to be compensated for the time value of money, or not having the use of that money while it is lent. In addition, they will want to be compensated for the expected value of the loss of purchasing power when the loan is repaid. These expected losses include the possibility that the borrower will default or be unable to pay on the originally agreed upon terms, or that collateral backing the loan will prove to be less valuable than estimated; the possibility of changes in taxation and regulatory changes which would prevent the lender from collecting on a loan or having to pay more in taxes on the amount repaid than originally estimated; and the loss of buying power compared to the money originally lent, due to inflation. :Nominal interest rates measure the sum of the compensations for all three sources of loss, plus the time value of the money itself.
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In making a bet where the expected value is positive, one is said to be getting "the best of it". For example, if one were to bet $1 at 10 to 1 odds (one could win $10) on the outcome of a coin flip, one would be getting "the best of it" and should always make the bet (assuming a rational and risk- neutral attitude with linear utility curves and have no preferences implying loss aversion or the like). However, if someone offered odds of 10 to 1 that a card chosen at random from a regular 52 card deck would be the ace of spades, one would be getting "the worst of it" because the chance is only 1 in 52 that the ace will be chosen. In an entry for L'Encyclopédie (the Enlightenment-era "French Encyclopedia"), Denis Diderot cites a similar example in which two players, Player A and Player B, wager over a game of dice that involves rolling two six-sided dice.
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It displays estimates on the y axis along with FIC scores on the x axis; thus estimates found to the left in the plot are associated with the better models and those found in the middle and to the right stem from models less or not adequate for the purpose of estimating the focus parameter in question. Generally speaking, complex models (with many parameters relative to sample size) tend to lead to estimators with small bias but high variance; more parsimonious models (with fewer parameters) typically yield estimators with larger bias but smaller variance. The FIC method balances the two desired data of having small bias and small variance in an optimal fashion. The main difficulty lies with the bias b_j , as it involves the distance from the expected value of the estimator to the true underlying quantity to be estimated, and the true data generating mechanism may lie outside each of the candidate models.
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The oldest and most common betting system is the martingale, or doubling-up, system on even-money bets, in which bets are doubled progressively after each loss until a win occurs. This system probably dates back to the invention of the roulette wheel. Two other well- known systems, also based on even-money bets, are the d’Alembert system (based on theorems of the French mathematician Jean Le Rond d’Alembert), in which the player increases his bets by one unit after each loss but decreases it by one unit after each win, and the Labouchere system (devised by the British politician Henry Du Pré Labouchere, although the basis for it was invented by the 18th-century French philosopher Marie-Jean-Antoine-Nicolas de Caritat, marquis de Condorcet), in which the player increases or decreases his bets according to a certain combination of numbers chosen in advance. The predicted average gain or loss is called expectation or expected value and is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value).
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Among the other contributions in this book, Whitworth was the first to use ordered Bell numbers to count the number of weak orderings of a set, in the 1886 edition. These numbers had been studied earlier by Arthur Cayley, but for a different problem.. He was the first to publish Bertrand's ballot theorem, in 1878; the theorem is misnamed after Joseph Louis François Bertrand, who rediscovered the same result in 1887.. He is the inventor of the E[X] notation for the expected value of a random variable X, still commonly in use,. and he coined the name "subfactorial" for the number of derangements of n items.. Another of Whitworth's contributions, in geometry, concerns equable shapes, shapes whose area has the same numerical value (with a different set of units) as their perimeter. As Whitworth showed with D. Biddle in 1904, there are exactly five equable triangles with integer sides: the two right triangles with side lengths (5,12,13) and (6,8,10), and the three triangles with side lengths (6,25,29), (7,15,20), and (9,10,17)..
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If the checksums match, the data are passed up the programming stack to the process that asked for it; if the values do not match, then ZFS can heal the data if the storage pool provides data redundancy (such as with internal mirroring), assuming that the copy of data is undamaged and with matching checksums. It is optionally possible to provide additional in-pool redundancy by specifying (or or more), which means that data will be stored twice (or three times) on the disk, effectively halving (or, for , reducing to one third) the storage capacity of the disk. Additionally some kinds of data used by ZFS to manage the pool are stored multiple times by default for safety, even with the default copies=1 setting. If other copies of the damaged data exist or can be reconstructed from checksums and parity data, ZFS will use a copy of the data (or recreate it via a RAID recovery mechanism), and recalculate the checksum—ideally resulting in the reproduction of the originally expected value.
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Second, a series of simulations show that fast-and-frugal trees with different exit structures will lead to different—sometimes drastically different—expected value of a decision when the consequences of a miss and a false alarm differ. Therefore, when constructing and applying a fast-and-frugal tree, one needs to choose an exit structure that matches well the decision payoff structure of a task. Third, the overall sensitivity of a fast-and-frugal tree—that is, how well the tree can discriminate a signal from a noise and which can be measured by d’ or A’ from signal detection theory—is affected by properties of the cues that make up the tree, such as the mean and variance of the cues’ sensitivities and the inter-cue correlations among the cues, but not much by the exit structure of the tree. And finally, the performance of fast-and- frugal trees is robust and comparable to much more sophisticated decision algorithms developed in signal detection theory, including the ideal observer analysis model and the optimal sequential sampling model.
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The results of these researchers shows that (in theory, assuming that the field of play is the infinite plane rather than a bounded rectangle) it is always possible to solve the puzzle while leaving n^\epsilon of the n input vertices fixed in place at their original positions, for a constant \epsilon that has not been determined precisely but lies between 1/4 and slightly less than 1/2. When the planar graph to be untangled is a cycle graph, a larger number of vertices may be fixed in place. However, determining the largest number of vertices that may be left in place for a particular input puzzle (or equivalently, the smallest number of moves needed to solve the puzzle) is NP-complete. has shown that the randomized circular layout used for the initial state of Planarity is nearly the worst possible in terms of its number of crossings: regardless of what planar graph is to be tangled, the expected value of the number of crossings for this layout is within a factor of three of the largest number of crossings among all layouts.
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Sebo calls the question of how to treat individuals of uncertain sentience, the "sentience problem" and argues that this problem which "Wallace raises deserves much more philosophical attention than it currently receives." Sebo asserts that there are two motivating assumptions behind the problem: "sentientism about moral status"—the idea that if an individual is sentient, that they deserve moral consideration—and "uncertainty about other minds", which refers to scientific and philosophical uncertainty about which individuals are sentient. In response to the problem, Sebo lays out three different potential approaches: the incautionary principle, which postulates that in cases of uncertainty about sentience it is morally permissible to treat individuals as if they are not sentient; the precautionary principle, which suggests that in such cases we have a moral obligation to treat them as if they are sentient; and the expected value principle, which asserts that we are "morally required to multiply our credence that they are by the amount of moral value they would have if they were, and to treat the product of this equation as the amount of moral value that they actually have". Sebo advocates for the latter position.
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As a background fact, we use the identity E[x^2] = \mu^2 + \sigma^2 which follows from the definition of the standard deviation and linearity of expectation. A very helpful observation is that for any distribution, the variance equals half the expected value of (x_1 - x_2)^2 when x_1, x_2 are an independent sample from that distribution. To prove this observation we will use that E[x_1x_2] = E[x_1]E[x_2] (which follows from the fact that they are independent) as well as linearity of expectation: :E[(x_1 - x_2)^2] = E[x_1^2] - E[2x_1x_2] + E[x_2^2] = (\sigma^2 + \mu^2) - 2\mu^2 + (\sigma^2 + \mu^2) = 2\sigma^2 Now that the observation is proven, it suffices to show that the expected squared difference of two observations from the sample population x_1, \ldots, x_n equals (n-1)/n times the expected squared difference of two observations from the original distribution. To see this, note that when we pick x_u and x_v via u, v being integers selected independently and uniformly from 1 to n, a fraction n/n^2 = 1/n of the time we will have u = v and therefore the sampled squared difference is zero independent of the original distribution.
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