I'm trying to solve the following problem:
Let $X$ & $Z$ be real valued r.v.'s with absolutely continuous joint distribution, whose density is given by:
$$f_{(X,Z)}= e^{-z}\mathbf{1}_{\{0\leq x \leq z\}}$$
Find the marginal densities of $X$ and $Z$.
I know the marginal densities are given by:
$$f_X = \int{f_{(X,Z)}}dz$$ $$f_Z = \int{f_{(X,Z)}}dx$$
My confusion is caused by the indicator function in the joint density. What are my boundaries for each integral? Do I have to "ignore" $x$ and integrate from $0$ to $z$ for $dz$? What do I do for $dx$?
Any help would be appreciated.