TITLE: Question about operations involving distributions
QUESTION [0 upvotes]: Having two independent and identically distributed distributions $X$~$N(0,1)$ and $Y$ I want to show that $U=\frac{XY}{\sqrt{X^2+Y^2}}$ is also a normal distribution and determine its variance.
Now I know $XY$ is a chi-squared distribution with one degree of freedom, and $X^2 + Y^2$ with two, but am having trouble taking the next step.

REPLY [0 votes]: Are you sure? Consider the fact that $XY$ is the product of two independent $N(0,1)$ r.v.; that means, for instance—since $XY<0 \iff (X<0\wedge Y>0) \vee (X>0\wedge Y<0)$—that 
$$P(XY<0)=P(X<0)P(Y>0)+P(X>0)P(Y<0))=\frac12 \cdot \frac12 + \frac12 \cdot \frac12=\frac12$$.
Is it possible then that $XY \sim \chi^2_1$? What's the prob of a $\chi^2_n$ r.v. taking negative values?
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