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Math Help - Expected Value Question

  1. #1
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    Expected Value Question

    Hello everyone,

    I have the following question. Suppose that X and Y are independent and f(x,y) is nonnegative. Put g(x)=E[f(x,Y)] and show E[g(X)]=E[f(X,Y)]. Show more generally that Integral over X in A of g(X) dP = Integral over X in A of f(X,Y) dP. Extend to f that may be negative. I've had no issues, except with the extension to negative f part. Any suggestions?
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  2. #2
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    Re: Expected Value Question

    Hello,

    I suggest you write the expectations with respect to which rv you consider it : E_X[m(X)]=\int m(x) ~dP_X for example.

    So we actually have g(x)=E_Y[f(x,Y)], we want to prove that E_X[g(X)]=E_{(X,Y)}[f(X,Y)]

    But E_X[g(X)]=E_X[E_Y[f(X,Y]]=E_X\left[\int_\Omega f(X,y) ~dP_Y\right]=\int_\Omega \int_\Omega f(x,y)~dP_Y dP_X

    And we also have E_{(X,Y)}[f(X,Y)]=\int_{\Omega^2} f(x,y)~dP_{(X,Y)}
    But since X and Y are independent, dP_{(X,Y)}=dP_X dP_Y
    So E_{(X,Y)}[f(X,Y)]=\int_\Omega \int_\Omega f(x,y) ~dP_Y dP_X
    (I think this is where the positiveness of f intervenes, in order to apply Fubini's theorem)

    And hence the equality.
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