I am provided the following joint density:
$$ f_{X,Y}(x,y)~=~\begin{cases} \lambda^3xe^{-\lambda y} & 0\le x \le y\\ \end{cases} $$
I am supposed to be able to find the density of $Y$ and $E(Y)$. Note that $\lambda$ is just a constant.
I have calculated the density of $Y$ as:
\begin{equation} \begin{split} f_Y(y) &= \int_0^y\lambda^3xe^{-\lambda y}dx \\ &= \frac{1}{2}\lambda^3y^2e^{-\lambda y} \end{split} \end{equation}
But I do not know what my bounds are on $f_Y(y)$. Would it be $x<y<\infty$ since $y$ is unbound in the positive direction in the joint density?
If so, then is the expectation of Y below correct?
$$ E(Y) = \int^{\infty}_0yf_Y(y)dy $$