Find the PDF of $U = {XY \over \sqrt{X^2+Y^2}}$ where $X, Y \sim N(0, 1)$ where X,Y are iid RV. Hence find the mean and variance of $U$.
Given that $X,Y$ are independent $N(0,1)$ , show that $\frac{XY}{\sqrt{X^2+Y^2}},\frac{X^2-Y^2}{2\sqrt{X^2+Y^2}}$ are independent $N(0,\frac{1}{4})$