Applied Probability Calculation not Working Out (Integrals)

[jumbogala]

Homework Statement

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.

Homework Equations

The Attempt at a Solution

First I need to find f_X|Y (x|y). To do this I need to take f(x,y) and divide by f_Y(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?

 

[fzero]

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?

The integral is

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