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Lecture 08-2007 - MEAN AND HIGHER MOMENTS Lecture VIII...

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MEAN AND HIGHER  MOMENTS  Lecture VIII 
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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 ]= Σ i =1 x i P ( x i ) if the series converges  absolutely.  We can write  E [ X ]= Σ + x i P ( x i )+  Σ - 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  Σ + x i P ( x i )=  and  Σ - x i P ( x i )=-  then  E [ X ] does  not exist.  
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If  Σ + x i P ( x i )=  and  Σ -  finite then we say  E [ X ]= .   If  Σ - x i P ( x i )=-  and  Σ +  is finite then we say that  E [ X ]=- .
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ROLLING DICE Number Probability E[X]= Σ 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
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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
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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:
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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 11.287 23.8282 50 0.215 10.7495 23.649 55 0.1209 6.6513 15.1165 60 0.0403 2.4186 5.6435 65 0.006 0.393 0.9372 45 95
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In the parlance of risk theory, the expected value  of the wheat crop is termed the actuarial or fair  value of the game.  It is the value that a risk  neutral individual would be willing to pay for the  bet. 
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