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Math Help - Conditional expected value

  1. #1
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    Conditional expected value

    Can someone show me how E[E[Y|X=x]]=E[Y]

    You can use your own joint distribution, or the uniform one I included below

    Conditional expected value-dyyyyf5sxw.png

    f(x,y)=\frac{2}{3}

    f_{Y|X}(y|X=x)=\frac{1}{2-y}\text{ for } y\leq x\leq 2
    Last edited by downthesun01; August 27th 2013 at 02:42 AM.
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  2. #2
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    Re: Conditional expected value

    Hey downthesun01.

    The general proof involves using the definitions (in continuous and discrete probability spaces) and basically "integrating out" the Y random variable after you integrate out the X random variable for a particular value.

    Law of total expectation - Wikipedia, the free encyclopedia

    For your particular case take the joint distribution, integrate out the x variable and then take the resulting function (which will depend on y) and integrate out the y variable.

    Also note that when you integrate out you need to multiply the PDF by the variable (since E[X] = Integral x*f(x)dx for some RV x).
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