Let $X$ and $Y$ be continuous random variables having joint density $f$ given by $f(x,y) = \lambda^2 e^{-\lambda y}, 0 \leq x \leq y$ and $f(x,y) = 0$ elsewhere.
Find the marginal densities of $X$ and $Y$. Find the joint distribution function of $X$ and $Y$.
This is what I have but I'm sure it is incorrect:
$f_X(x) = \int_0^{\infty} f(x,y) dy = \int_0^{\infty} \lambda^2 e^{-\lambda y} dy = \lambda(\lim_{y \rightarrow \infty} e^{-\lambda y} - e^{-\lambda 0}) = \lambda$