Question
Suppose that random variables $X$ and $Y$ have joint probability density function (PDF) $$f_{X, Y}(x, y) = e^{-x},\quad 0 < y < x.$$
Find $\mathbb{E}(Ye^{-X})$ and $\mathbb{E}(Ye^{-X}\ |\ Y = 2)$.
My working
$$\mathbb{E}(Ye^{-X}) = \int^{\infty}_y \int^x_0 ye^{-x}f_{X, Y}(x, y)\ \mathrm{d}y\ \mathrm{d}x$$
$$\mathbb{E}(Ye^{-X}\ |\ Y = 2) = \int^{\infty}_0 2e^{-x}f_{X|2}(x|2)\ \mathrm{d}x$$
I have just covered (conditional) expectations for joint random variables and would like to know if my methods for calculating them are correct. In particular, are my limits of integration for the respective integrals right? If not, please do tell where I have gone wrong :)