Deriving joint and marginal PDFs

Question

If $R$ is an exponential random variable with parameter 1, $\theta$ is uniform $[0, 2\pi]$, and

$A = \sqrt{R}cos(\theta)$, $B = \sqrt{R}sin(\theta)$

How would I find the joint PDF of A and B and the marginal PDF of A?

Answer 1

It appears that $R=A^2+B^2$ and $\theta = {\arctan(B/A)\cdot\mathbf 1_{A\neq 0, B>0}\\+(2\pi-\arctan(-B/A))\cdot\mathbf 1_{A\neq 0, B<0}\\+\pi\cdot\mathbf 1_{A<0,B=0}}$

Now, $\left\lVert\dfrac{\partial(x^2+y^2, \arctan(y/x))}{\partial (x,y)}\right\rVert = \begin{Vmatrix}2x & \frac{-y}{x^2+y^2}\\ 2y & \frac{x}{x^2+y^2} \end{Vmatrix} = 2$

And, $\left\lVert\dfrac{\partial(x^2+y^2, 2\pi-\arctan(-y/x))}{\partial (x,y)}\right\rVert = \begin{Vmatrix}2x & \frac{-y}{x^2+y^2}\\ 2y & \frac{x}{x^2+y^2} \end{Vmatrix} = 2$

So use this to apply the Jacobian change of variables theorem.