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Lecture 08-2007

Lecture 08-2007 - Mean and Higher Moments Lecture VII I...

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Mean and Higher Moments Lecture VII I. Expected Value A. Definition 4.1.1. Let X be a discrete random variable taking the value i x with probability i Px , 1,2, i . Then the expected value ( expectation or mean ) of X , denoted EX , is defined to be 1 ii i E X x P x if the series converges absolutely. We can write i i i i E X x P x x P x where in the first summation we sum for i such that 0 i x and in the second summation we sum for i such that 0 i x . If x P x and x P x then Ex does not exist. If x P x and x P x is finite then we say . If x P x and x P x is finite then we say that . B. Taking a practical application: 1. Given that each face of the die is equally likely, what is the expected value of the role of the die? Table 1. Expected Value of a Single Die Role Number Probability 1 i E X x P x 1 0.167 0.167 2 0.167 0.333 3 0.167 0.500 4 0.167 0.667 5 0.167 0.833 6 0.167 1.000 3.500 2. What is the expected value of a two-die role?

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AEB 6933 Mathematical Statistics for Food and Resource Economics Lecture VIII Professor Charles B. Moss Fall 2007 2 Table 2. Expected Value of the Role of Two Die Die 1 Die 2 Number 1 ii i E X x P x Die 1 Die 2 Number 1 i E X x P x 1 1 2 0.056 1 4 5 0.139 2 1 3 0.083 2 4 6 0.167 3 1 4 0.111 3 4 7 0.194 4 1 5 0.139 4 4 8 0.222 5 1 6 0.167 5 4 9 0.250 6 1 7 0.194 6 4 10 0.278 1 2 3 0.083 1 5 6 0.167 2 2 4 0.111 2 5 7 0.194 3 2 5 0.139 3 5 8 0.222 4 2 6 0.167 4 5 9 0.250 5 2 7 0.194 5 5 10 0.278 6 2 8 0.222 6 5 11 0.306 1 3 4 0.111 1 6 7 0.194 2 3 5 0.139 2 6 8 0.222 3 3 6 0.167 3 6 9 0.250 4 3 7 0.194 4 6 10 0.278 5 3 8 0.222 5 6 11 0.306 6 3 9 0.250 6 6 12 0.333 7.000 C. Expectation has several applications in risk theory. In general, the expected value is the value we expect to occur. For example, if we assume that the crop yield follows a binomial distribution as depicted in figure 1, the expected return on the
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Lecture 08-2007 - Mean and Higher Moments Lecture VII I...

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