Assume $U$ is uniform on $(0, 2\pi)$ and $Z$, independent of $U$, is exponential with rate $\lambda=1$. Now assume $X$ and $Y$ are defined by: $X = \sqrt{2Z} \cos(U)$ and $Y = \sqrt{2Z}\sin(U)$. Find the joint distribution of $X$ and $Y$.
I am stuck on this question while reviewing for my test next week.
I assume that I need to manipulate $X$ and $Y$ in terms of $U$ and $Z$, since I am given the fact that $U$ and $Z$ are independent, and therefore $f(U,Z) = f(U)f(Z)$. However the question asks about $f(X,Y)$. I'm pretty sure that I can't just substitute the distributions of $U$ and $Z$ in the equations of $X$ and $Y$, but I'm not sure how I can solve this?
Thanks!
HINT
One possible approach consists in applying the change of variables method. To begin with, notice that
\begin{cases} U = \displaystyle\arctan\left(\frac{Y}{X}\right)\\\\ Z = \displaystyle\frac{X^{2} + Y^{2}}{2} \end{cases}
Therefore we have
\begin{align*} f_{X,Y}(x,y) & = f_{U,Z}\left(\arctan\left(\frac{y}{x}\right),\frac{x^{2}+y^{2}}{2}\right)|\det J(x,y)|\\ & = f_{U}\left(\arctan\left(\frac{y}{x}\right)\right)f_{Z}\left(\frac{x^{2}+y^{2}}{2}\right)|\det J(x,y)| \end{align*}
since $U$ and $Z$ are independent. Can you proceed from here?