A truly practical math curriculum would have game theory, graph theory, mathematical expectation and proof deeply installed as early as third grade.
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Lee Erickson, a critic, has compared the book's conclusion to John Maynard Keynes's view of the economic importance of "spontaneous optimism, rather than mathematical expectation".
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In such games, the terminal time is a random variable with a given probability distribution function. Therefore, the players maximize the mathematical expectation of the cost function. It was shown that the modified optimization problem can be reformulated as a discounted differential game over an infinite time interval.
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There are several problems with the assumptions that Due-column bettors make, not the least of which is that the mathematical basis for the system is flawed. Assuming a bettor in any series of horse races is a 33 percent handicapper with an average mutuel of $5.00. Such a bettor's mathematical expectation can be expressed as follows: E(x) = .33 <\--> ($1.50) - .
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For example: mathematical expectation of the efficiency; the total probability of assuring preset constraints; variance of the efficiency and so on It is evident that the extreme of the one of these criteria doesn't guarantee the assurance of the high level of another one. Even more, these criteria may contradict to each other. Thus, in this case we have a multiobjective optimization problem.
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67 = -$.175 According to this analysis, the average bettor's mathematical expectation would not provide a profit, but a loss because his expectation is a negative gain every bet. As a result, the bettor's playing the Due-column system will only multiply his losses over time. After 10 losses given the expression above, for instance, the bettor would average a loss of $1.75, thus making an upward-sloping expectation impossible.
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Therefore, random weight-vector (W(1;I),…,W(m;I)) induces randomization of an aggregated index Q, i.e., its transformation in the corresponding randomized aggregated index Q(I). The looked for average aggregated estimation of objects’ quality level may be identified now with mathematical expectation of corresponded random aggregated index Q(I). The measure of the aggregated estimation’s exactness may be identified with the standard deviation of the correspondent random index.
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Both Chuprov and his student Oskar Anderson published in Pearson's journal Biometrika. Chuprov was not above telling the English off, "English scientific tradition rejects the concept of 'mathematical probability' ... and the method of mathematical expectation has naturally shared the fate of the concept ... on which it rests." For a brief period Chuprov was known in Britain. In John Maynard Keynes' Treatise on Probability (1921) he is put with Markov and Chebyshev as the three great Russian names in the theory of statistics.
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This type of indices involves choices under risk. In this case, , and , are lotteries associated with outcomes. Unlike cardinal utility theory under certainty, in which the possibility of moving from preferences to quantified utility was almost trivial, here it is paramount to be able to map preferences into the set of real numbers, so that the operation of mathematical expectation can be executed. Once the mapping is done, the introduction of additional assumptions would result in a consistent behavior of people regarding fair bets.
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There are interesting thoughts on what probability really is: > ... probability as a measurable degree of certainty; necessity and chance; > moral versus mathematical expectation; a priori an a posteriori probability; > expectation of winning when players are divided according to dexterity; > regard of all available arguments, their valuation, and their calculable > evaluation; law of large numbers ... Bernoulli was one of the most significant promoters of the formal methods of higher analysis. Astuteness and elegance are seldom found in his method of presentation and expression, but there is a maximum of integrity.
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Bernoulli assumed that "a poor man generally obtains more utility than a rich man from an equal gain" an approach that is more profound than the simple mathematical expectation of money as it involves a law of moral expectation. Early theorists of utility considered that it had physically quantifiable attributes. They thought that utility behaved like the magnitudes of distance or time, in which the simple use of a ruler or stopwatch resulted in a distinguishable measure. "Utils" was the name actually given to the units in a utility scale.
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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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In economics, this circle of ideas is analysed under the rubric of "Knightian uncertainty". John Maynard Keynes and Frank Knight both discussed the inherent unpredictability of economic systems in their work and used it to criticise the mathematical approach to economics, in terms of expected utility, developed by Ludwig von Mises and others. Keynes in particular argued that economic systems did not automatically tend to the equilibrium of full employment owing to their agents' inability to predict the future. As he remarked in The General Theory of Employment, Interest and Money: ... as living and moving beings, we are forced to act ... [even when] our existing knowledge does not provide a sufficient basis for a calculated mathematical expectation.
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In 2016 -2017 he formulated the so-called "Gafurov postulate" of political science on Russian TV. According to this hypothesis, in any election "from the United States to the DPRK, from India to Russia, from Switzerland to the People's China, from Germany to Syria," the number of illegal, "thrown in" votes is approximately the same, in cases of one or two candidates ranging from 5% to 10% of votes. Gafurov assumes the existence of a stable mathematical expectation of this figure in the presence of a sufficiently large number of polling stations and a large number of elections. "Gafurov's Political Science Hypothesis" is used to calculate the budgets of election campaigns in Russia. Gafurov had struggled to support the Syrian government, but after direct involvement of Russian forces in the Syrian conflict, he started to speak against the main line of Russian propaganda.
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In accordance with the concept, an uncertain choice of a weight-vector from set W(I) is modeling by a random choice of an element of the set. Such randomization produces a random weight- vector W(I)=(W(1;I),…,W(m;I)), which is uniformly distributed on the set W(I). Mathematical expectation of random weight-coefficient W(i;I) may be used as a numerical estimation of particular index (criterion) q(i) significance, exactness of this estimation being measured by standard deviation of the corresponding random variable. Since such estimations of single indices significance are determined on the base of NNN-information I, these estimations may be treated as a result of quantification of the non-numerical, inexact and incomplete information I. An aggregative function Q(q(1),…,q(m)) depends on weight-coefficients.
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