Let $X, Y$ be independent randoLet$U, V$ be independent of $X,Y$ .Show that $Z=\frac{(UX+VY)}{(U^2+V^2)^{\frac12}} \sim N(0,1)$

Question

Let $X, Y$ be independent random variables $N (0, 1)$ . Let $U, V$ be independent of $X, Y$ .Show that $Z = \frac{(UX + VY)}{(U^2 + V^2)^{\frac 12}} \sim N (0, 1)$

Answer 1

You can compute the characteristic function : \begin{align*} \mathbb E[e^{itZ}] &= \mathbb E\left[\mathbb E\left[e^{it\frac{UX+VY}{\sqrt{U^2+V^2}}}\middle|U,V\right]\right]\\ &= \mathbb E\left[\mathbb E\left[e^{it\frac{U}{\sqrt{U^2+V^2}}X}\middle|U,V\right]\mathbb E\left[e^{it\frac{V}{\sqrt{U^2+V^2}}Y}\middle|U,V\right]\right]\\ &= \mathbb E\left[e^{-\frac{1}{2}t^2\left(\frac{U}{\sqrt{U^2+V^2}}\right)^2}e^{-\frac{1}{2}t^2\left(\frac{V}{\sqrt{U^2+V^2}}\right)^2}\right]\\ &= e^{-\frac{1}{2}t^2} \end{align*} Using the conditional characteristic function.