47. Let $X$ and $Y$ be independent random variables with distribution $\mathscr{N}(0,1)$. Define

$$
T=\frac{X^{2}+Y^{2}}{2}, \quad g=\frac{X^{2}}{X^{2}+Y^{2}}
$$

Using problem II.8.11, find the joint distribution of $(T, g)$.
Solution. Due to problem II.8.11, the quantities $T$ and $g$ are independent, so it is sufficient to compute the distribution functions of these quantities. The quantity $T$ has an exponential distribution $\mathrm{P}(T>t)=e^{-t}, t>0$. The quantity $g$ has an arcsin distribution with density $\frac{1}{\pi \sqrt{x(1-x)}}, x \in(0,1)$.