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Density function of random variables E(X|Y) and E(Y|X)

  1. Feb 17, 2010 #1
    1. The problem statement, all variables and given/known data
    Let X and Y have JD f(x,y) = e^-y, 0<x<y

    Find:
    a) E(X|Y=y), E(Y|X=x)
    b) density function of R.V. E(X|Y), E(Y|X)



    3. The attempt at a solution
    a)
    I have found E(X|Y=y) = y/2 for y>= 0

    E(Y|X=x) = x +1 for x>= 0
    by finding fx(x) = ∫(x to infinity) e^-y dy = e^-x
    f(y|x) = (e^-y)/ (e^-x)
    so E(Y|X=x) = ∫( x to inifinty) y* (e^-y)/ (e^-x) dy = x +1

    I was wondering if u actually take the integral from x to INFINITY since the RESTRICTION is 0<x<y.

    b) E(X|Y=y) = Y/2 for y>= 0

    E(Y|X=x) = X +1 for x>= 0

    I was wondering for this question if you just convert the x and y to capital letters? or if i am suppose to do something else
     
  2. jcsd
  3. Feb 17, 2010 #2
    a) That's right.
    b) Basically correct, I'm assuming you mean find E(X|Y), which is simply E(X|Y=y) evaluated at y = Y.
     
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