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Math Help - Find E[X] and E[E[X|Y]]

  1. #1
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    Find E[X] and E[E[X|Y]]

    Let X and Y be random variables with  p(x,y)=e^ {-y} for  0<x<y<+ \infty

    Find  E[X] and  E[E[X|Y]]

    My solution:

    a) First of all,  p(x) = e^{-x} , so I have  E[X] = \int ^ y _0 xe^{-x}dx = -ye^{-y}-e^{-y}+1

    b) Here,  E[X|Y] = \frac {Y}{2} and  p(y)=ye^{-y} , so I have  E[E[X|Y]] = \int E[X|Y=y]p(y)dy = \int ^ \infty _x \frac {y}{2}(ye^{-y})dy= \frac {1}{2} e^{-x}(x^2+2x+2)

    But according to the law of total expectation, they should be the same thing, so what am I doing wrong? Thank you!
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  2. #2
    MHF Contributor matheagle's Avatar
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    Re: Find E[X] and E[E[X|Y]]

    part of this was asked in another question here

    (a) Yes, f_X(x)=e^{-x}
    BUT that is the marginal, hence the range is x>0
    you no longer want x<y.
    Plus the E(X) is a constant, which cannot have Y's in it.
    E(X)=1, this is an exponential

    (b) Yes, E(X|Y)=y/2

    and f_Y(y)=ye^{-y}, y>0 a Gamma RV

    So E_Y(E(X|Y)) =\int_0^{\infty}(y/2)ye^{-y}dy=(1/2)\int_0^{\infty}y^2e^{-y}dy=(1/2)\Gamma(3)=1.
    Last edited by matheagle; September 5th 2011 at 07:08 AM.
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