$(4)(2$ Points$)$ Let $U$ be $U(0,2)-$distributed, $R$ be independent of $U$ with distri$-$

bution function

$F_R(r)=\{\begin{array}{ccc}1-\text{e}\text{x}\text{p}& (-\frac{r^2}{2}),& r0\\ 0& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \end{array}$

Show that $(x,Y)$:$=g(R,),$ where $g$ is the function given in $(3),$ is $N(0,)-$

distributed with $=([1,0],[0,1]).$ Also, show that $R=\sqrt[\phantom{2}]{x^2+Y^2}$ and find

$E[\sqrt[\phantom{2}]{x^2+Y^2}]$

Hint: ${\displaystyle \underset{0}{\overset{}{}}}{r}^2{e}^{-\frac{r^2}{2}}dr=\sqrt[\phantom{2}]{\frac{}{2}}$