Lecture08-2010 - Mean and Higher Moments Charles B Moss July 9 2010 I Expected Value A Denition 4.1.1 Let X be a discrete random variable taking

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Mean and Higher Moments Charles B. Moss July 9, 2010 I. Expected Value A. Defnition 4.1.1 :Le t X be a discrete random variable taking the value x with probability P ( x i ) ,i =1 , 2 , ··· . Then the expected value (expectation or mean) of X , denoted E [X], is deFned to be E[X]= i=1 x i P(x i ) if the series converges absolutely. 1. We can write E [X] = + x i i )+ x i i ) where in the Frst summation we sum for i such that x i > 0 and in the second summation we sum for i such that x i < 0. 2. If + x i P ( x i )= and x i P ( x i then E [X] does not exist. 3. If + x i P ( x i and x i P ( x i ) is Fnite then we say . 4. If P ( x i −∞ and + x i P ( x i ) is Fnite then we say that −∞ . B. 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? 2. What is the expected value of a two-die role? C. Expectation has several applications in risk theory. In general, the expected value is the value we expect to occur. ±or example, if we assume that the crop yield follows a binomial distribution as depicted in ±igure 1, the expected return on the crop given that the price is $3 and the cost per acre is $40, becomes $ 95 per acre as demonstrated in Table 3. 1
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AEB 6571 Econometric Methods I Professor Charles B. Moss Lecture VIII Fall 2010 Table 1: Expected Value of a Single Die Role Number Probability x i P ( x i ) 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 Total 3.500 Table 2: Expected Value of a Two-Die Role Die 1 Die 2 Number x i P ( x i ) Die 1 Die 2 Number x i P ( x i ) 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 Total 7.000 2
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AEB 6571 Econometric Methods I Professor Charles B. Moss
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This note was uploaded on 07/15/2011 for the course AEB 6180 taught by Professor Staff during the Spring '10 term at University of Florida.

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Lecture08-2010 - Mean and Higher Moments Charles B Moss July 9 2010 I Expected Value A Denition 4.1.1 Let X be a discrete random variable taking

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