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**Crescent** I ran into this problem:

Suppose $\displaystyle X, Y IID N(0,1)$, obtain the pdf for $\displaystyle U=\frac{X}{\sqrt{X^2+Y^2}}$.

Because of the nasty square root, I can't seem to find the appropriate transformation functions for a change of variables. I tried letting $\displaystyle R=\sqrt{X^2+Y^2}$ and looking for the joint pdf of $\displaystyle U$ and $\displaystyle R$, but the function is not 1-1. Since $\displaystyle Z=\frac{Y}{X}$ is a standard cauchy, I've also tried breaking $\displaystyle U$ into: $\displaystyle -\frac{1}{\sqrt{1+Z^2}}$ for $\displaystyle x<0, 0$ for $\displaystyle x=0, \frac{1}{\sqrt{1+Z^2}}$ for $\displaystyle x>0$, but that doesn't help much.

What am I supposed to do when I see this type of questions? I need some hints.