# Probability: Conditional expectation from a joint pdf

Let $$X$$ and $$Y$$ have the joint pdf $$f(x;y)=2\exp(x+y)$$, $$0 < x < y < \infty$$, zero elsewhere. Find the conditional mean $$E(Y|X=x)$$.

This seems like a simple problem. I know I have to find the marginal pdf of $$x$$ and then divide the joint pdf by the marginal pdf. But the marginal pdf of $$X$$ diverges, so I don't know how to proceed.

$$\int_0^\infty \int_x^\infty 2\exp(x+y) \,\, dy dx \ne 1.$$
If we fix an $x$, $y$ can get arbitrarily large, and the so call pdf can get arbitraily large and it can't converges.
$$\int_0^\infty \int_x^\infty 2\exp(\color{red}-(x+y)) \,\, dy dx = 1.$$