Lecture05-2010 - Expected Value and Moments Charles B. Moss...

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Expected Value and Moments Charles B. Moss August 28, 2010 I. Expected Value A. The random variable can also be described using a statistic. 1. One basic statistic encountered by students in statistics courses is the mean of a random variable. 2. Defnition 2.12 p-28 The expected value (expectation or mean) of a discrete random variable X , denoted E [ X ], is deFned as E [ X ]= X x i X x i P [ x i ]( 1 ) 3. Defnition 2.13 p-28 The expected value of a continuous random variable X is then deFned as E [ X Z x X xf ( x ) dx (2) 4. Taking the die roll as an example, if we let each side be equally likely the expected value of the roll of a fair die is E [ x 6 X i =1 iP [ i ]=1 1 6 +2 1 6 +3 1 6 +4 1 6 +5 1 6 +6 1 6 =3 1 2 (3) 6. This result points out an interesting fact about the expected value, namely that the expected value need not be an element of the sample set. 1
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AEB 6182 Agricultural Risk Analysis and Decision Making Professor Charles B. Moss Lecture V Fall 2010 7. Suppose we weight the die so that it is no longer fair. Specif- ically, assume that P [ i ]=1 / 9for i = { 1 , 2 , 5 , 6 } , P [3] = 3 / 9, and P [4] = 2 / 9 , then the expected value becomes E [ X ]= 6 X i =1 iP [ i 1 9 +2
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Lecture05-2010 - Expected Value and Moments Charles B. Moss...

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