Density of $(X,Y)$ when $X:=\sqrt{-2\log U}\cos(2\pi V)\\Y:=\sqrt{-2 \log U}\sin(2\pi V)$ $U,V \sim(0,1)$

Question

Let $U,V \sim(0,1)$ be two independent uniformly distributed random variables:

$$X:=\sqrt{-2\log U}\cos(2\pi V)\\Y:=\sqrt{-2 \log U}\sin(2\pi V)$$

How can I determine the density of the distribution of $(X,Y)$?

I know that $\frac{Y}{X}=\tan(2\pi V)$ and $\frac{X^2}{\log U}+\frac{Y^2}{\log U}=-2$ but I don't know if this helps here.

Answer 1

You mean $U,V$ are uniform, and so $(X,Y)$ are $N(0,1)$ in the margins, and clearly uncorrelated. That does not, however, prove them to be independent (though actually they are). To find the density properly, use the transformation law, by finding the Jacobian.

Answer 2

Using the relations you mentioned, one can solve for $U$ and $V$ in terms of $X$ and $Y$. Let's say $U=g(X,Y)$ and $V=h(X,Y)$.

Then, by the theorem of change of variables, we have $$f_{XY}(x,y)=f_{UV}(g(x,y),h(x,y))\cdot J(x,y),$$ where $f_{UV}$ is the joint PDF of $U$ and $V$, and so $$f_{UV}(u,v)=\left\{\begin{matrix}1& 0<u<1 \wedge 0<v<1 \\0 & \text{otherwise}\\ \end{matrix}\right.,$$ and $J$ is the Jacobian determinant of $u$ and $v$ with respect to $x$ and $y$, that is $$J(x,y)=\left|\begin{matrix} \frac{\partial g}{\partial x}(x,y)& \frac{\partial g}{\partial y}(x,y)\\\frac{\partial h}{\partial x}(x,y) & \frac{\partial h}{\partial y}(x,y)\\ \end{matrix}\right|.$$