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Math Help - Quick question on conditional expectation

  1. #1
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    Quick question on conditional expectation

    Let x,y be continous random variables.

    Prove that E(E(y|x)) = E(y)

    I am sure this is a very basic question but still if someone can provide/guide towards a rigourous proof for the same?

    Thanks
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  2. #2
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    In fact I realised that the more generic result will hold true

    E(E(f(x,y)|x)) = E(f(x,y)), where f(x,y) is a real valued function on (x,y)

    E stands for expected value.

    Any help/pointers to prove this (in a rigorous way) will be great ! Thanks
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  3. #3
    MHF Contributor matheagle's Avatar
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    It's true in any case, but the continuous case is easy...

    E(Y|X)=\int yf(y|X)dy which is a function of X

    next integrate wrt x and multiply the densities...

    E(E(Y|X))=\int\left(\int yf(y|x)dy\right)f_X(x)dx

    Fubini...  =\int\int yf(y|x)dyf_X(x)dx=\int\int yf(x,y)dxdy=E(Y)
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