# Conditional Joint Distributions

I am given random variables x and y where $$f(x,y) = 1$$ for $$0 <= x, y <= 1$$ and 0 elsewhere. I am asked to find $$E(X | Y > X)$$.

Here’s what I have. I know I made a mistake somewhere or am misunderstanding something so any help would be appreciated.

I get $$f_{x|y}(x, y > frac{1}{2}) = frac{f_{xy}(x, y > x)}{f_{y}(y > x)}$$.

I get $$f_{xy}(x, y > x) = int_{0}^{1} int_{x}^{1} 1 {,dy} {,dx} = int_{0}^{1} (1-x) {,dx} = 1/2$$

I also get $${f_{y}(y > x)} = int_{x}^{1} 1 {,dy} = 1-x$$

However, that means $$f_{x|y}(x, y > frac{1}{2}) = frac{1}{2(1-x)}$$.

This can’t be right since when I plug this into $$E(X | Y > X) = int_{0}^{1} x * frac{1}{2(1-x)} {,dx}$$, this is an undefined value…so clearly I messed up. What step am I doing wrong? Or am I not even approaching the problem in the right way?

Thanks!