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Homework Help: Applied Probability Calculation not Working Out! (Integrals)

  1. Sep 27, 2010 #1
    1. The problem statement, all variables and given/known data
    The joint density of X and Y is given by
    f(x,y) = [e-x/y)e-y ] / y

    x is between 0 and infinity; y is between 0 and infinity

    Show E[X |Y = y] = y.

    Jump to the last part of this post in bold if you just want to check my calculus calculations and skip the probability stuff.

    2. Relevant equations

    3. The attempt at a solution
    First I need to find fX|Y (x|y). To do this I need to take f(x,y) and divide by fY(y).

    I think it's the latter which is messing me up. My understanding is that to find it, I need to integrate f(x,y) with respect to x.

    The solutions manual does that, but there are no limits on their integral. Shouldn't the limits be from 0 to infinity?

    They have [(1/y)e-x/ye-y ] / [e-y ∫ (1/y)e(-x/y) dx ] = (1/y)e-x/y

    Which in my calculations doesn't work out! I get -1/y. What am I doing wrong?
  2. jcsd
  3. Sep 27, 2010 #2


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    Science Advisor
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    The integral is

    [tex] \int_0^\infty e^{-x/y} dx = -y ( 0 - 1 ) = y.[/tex]
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