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Math Help - Variance

  1. #1
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    Variance

    A company has a product liability lawsuit. It can settle out of court for 1.2 Million.
    If it elects to go to court the jury has a 75% of finding the company liable.
    If liable, the company will pay either 1 or 4 Million with probability .3 and 2 Million with probability .4.
    The Expected cost of going to court is therefore .75[1M(.3)+2M(.4)+4M(.3)]+(.25*0)=1.725M
    In order to compute the variance, I calculated the second moment as follows
    .75[(1M)^2*.3+(2M)^2*.4+(4M)^2*.3)]=5.025*10^12
    So the Variance of the cost of going to court would be 5.025*10^12-(1.725M)^2=2.049375*10^12, and the standard deviation would be $1,431,564. I'm
    fairly certain this is correct, however, I am curious as to how the Variance would have been calculated using the second part of THE DOUBLE EXPECTATION THEOREM, that is,
    VAR(X)=E[VAR(X|Y)]+VAR[E(X|Y)].
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  2. #2
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    Re: Variance

    Var(X)=E[Var(X|Y)]+Var[E(X|Y)]=.75(1.41*10^12)+(2.3M-0)^2(.25)(.75)=2.049375*10^12=
    (1.0575+.991875)*10^12=2.049375*10^12
    standard deviation= sqrt(2.049375*10^12)=$1,431,564 (to the nearest dollar)
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