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

[saizen21]

Homework Statement

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)

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

 

[Mandark]

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.