Nonlinear Transformations of Standard Normal iid Random Variables

Question

I'm having a rather difficult time with these types of problems.

Say we are are considering two random variables $X_1$, $X_2$ ~ $N(0,1)$.

We want to compute the distribution of $U=\frac{X_1}{X_1 + X_2}$.

My first attempts at this sought to use the the joint density of $U$, $V=X_2$ and then compute the marginal density. This brings us to a seemingly unsolvable integral. Is there some way to compute this without performing a wildly difficult gaussian integral?

Answer 1

Hints:

• $U=\frac{1}{1 + Y}$ where $Y=\frac{X_2}{X_1}$
• Consider the distribution of $\frac1U$
• If $X_1,X_2$ i.i.d. $N(0,1)$ then $\frac{X_2}{X_1} \sim \operatorname{Cauchy}(0,1)$
• If $Z \sim \operatorname{Cauchy}(x_0,\gamma)$ then $\frac1Z \sim \operatorname{Cauchy}\left(\frac{x_0}{x_0^2+\gamma^2},\frac{\gamma}{x_0^2+\gamma^2}\right)$