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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.

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There is probably a mistake in the question.

In fact, note that

$$\int_0^\infty \int_x^\infty 2\exp(x+y) \,\, dy dx \ne 1.$$

If we fix an $x$, $y$ can get arbitrarily large, and the so call pdf can get arbitraily large and it can't converges.

We do have

$$\int_0^\infty \int_x^\infty 2\exp(\color{red}-(x+y)) \,\, dy dx = 1.$$

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