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Unformatted text preview: MEAN AND HIGHER MOMENTS Lecture VIII EXPECTED VALUE Definition 4.1.1. Let X be a discrete random variable taking the value x i with probability P ( x i ), i =1,2,. Then the expected value E [ X ] is defined to be E [ X ]= i =1 x i P ( x i ) if the series converges absolutely. We can write E [ X ]= + x i P ( x i )+  x i P ( x i ) where in the first summation we sum for i such that x i >0 and in the second summation we sum for i such that x i <0. If + x i P ( x i )= and  x i P ( x i )= then E [ X ] does not exist. If + x i P ( x i )= and  finite then we say E [ X ]= . If  x i P ( x i )= and + is finite then we say that E [ X ]= . ROLLING DICE Number Probability E[X]= i=1 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 3.500 EXPECTED VALUE OF TWO DIE Die 1 Die 2 Number E[ X ]= 1 1 2 0.056 2 1 3 0.083 3 1 4 0.111 4 1 5 0.139 5 1 6 0.167 6 1 7 0.194 7 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 crop given that the price is $3 and the cost per acre is $40, becomes: EXPECTED PROFIT ON CROP 15 0.0001 0.0016 0.0005 20 0.0016 0.0315 0.0315 25 0.0106 0.2654 0.3716 30 0.0425 1.274 2.1234 35 0.1115 3.9017 7.246 40 0.2007 8.0263 16.0526 45 0.2508 11.28711....
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 Fall '09
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