Calculating conditional expectation, confused on integration bounds? I am given a joint density, $$f_{X,Y}(x,y)=a^2e^{-ay}$$I need to calculate both $E[X\mid Y=y]$ and $\operatorname{Var}[X\mid Y=y]$.
I am also given that $a>0$ and that $(0<x<y<\infty)$.

$$E[X\mid Y=y]=\int_{-\infty}^{\infty}xf_{X\mid Y}(x\mid y)\,dx$$
So I'll start by calculating marginal and conditional densities. But this is where I'm getting hung up on the bounds. I'll be integrating with respect to $y$ for this first integration, and I can't figure out what I should change the bounds to.
$$f_X(x)=\int_{-\infty}^{\infty}f_{X,Y}(x,y)dy$$
$$=\int_{-\infty}^{\infty}a^2e^{-ay}dy$$
Once I figure out what the bounds are, then I can calculate:
$$f_{X\mid Y}(x\mid y)=\frac{f(x,y)}{f_Y(y)}$$
And then, finally:
$$E[X\mid Y=y]=\int_{-\infty}^{\infty}xf_{X\mid Y}(x\mid y)\,dx$$
$$\operatorname{Var}[X\mid Y=y]=E[X^2\mid Y=y]-(E[X\mid Y=y])^2$$
So I generally see what to do here, but I am missing the really basic concept of the bounds. Can someone help?

\begin{align}
f_X(x)=\int_{-\infty}^{\infty}f_{X,Y}(x,y) \, dy & =\overbrace{\int_x^\infty a^2e^{-ay} \, dy} \\[10pt]
& = \left.-ae^{-ay}\vphantom{\frac11} \right|_{y\,:=\,x}^{y\,:=\,\infty} = ae^{-ax} & \text{for } x>0.
\end{align}
You had
$$
f_X(x)= \overbrace{\int_{-\infty}^\infty f_{X,Y}(x,y) \, dy}^\text{right} =\overbrace{\int_{-\infty}^\infty a^2e^{-ay} \, dy}^\text{wrong}.
$$
You can use this to find the marginal expected value and the marginal variance of $X$.
Where you wrote "I am given$\,\ldots$", you didn't finish the sentence until later where you wrote "I am also given$\,\ldots$".
