Q:

Finding PDF of $Z \thicksim Q / \sqrt{Q^2 + R^2}$ where $Q, R \thicksim N(0,1)$?

I have only done simple convolutions up to this point to find joint PDFs and am lost about how to proceed with finding the PDF of $Z \thicksim \frac{Q}{\sqrt{Q^2 + R^2}}$ where $Q, R \thicksim N(0,1)$. How do I approach finding the PDF of $Z$? 

A:

I assume that $Q$ and $R$ are independent. First, $F_Z(z)=0$ for $z\le -1$ and $F_Z(z)=1$ for $z\ge 1$. Also
$$
\mathsf{P}(Z\le z\)=\mathsf{E}\!\left[\mathsf{P}\!\left(Q\le z\sqrt{Q^2+R^2}\mid R\right)\right].
$$
For $z\in (-1,1)$,
$$
\mathsf{P}\!\left(Q\le z\sqrt{Q^2+r^2}\right)=\mathsf{P}\!\left(Q\le z\sqrt{\frac{r^2}{1-z^2}}\right)=\Phi\!\left(z\sqrt{\frac{r^2}{1-z^2}}\right)
$$
so that 
$$
F_Z(z)=\int_{-\infty}^{\infty}\Phi\left(z\sqrt{\frac{r^2}{1-z^2}}\right)\phi(r)dr=\frac{1}{2}+\frac{\arcsin(z)}{\pi}, \quad z\in (-1,1).
$$
and the corresponding pdf is $f_Z(z)=\frac{1}{\pi\sqrt{1-z^2}}$ on $(-1,1)$.

