<s>[INST] PDF of $Z=\frac{X^2+Y^2}{2}$ where $X\sim N(0,1)$ and $Y\sim N(0,1)$ Say $X \sim N(0,1)$ and $Y\sim N(0,1)$ are independent random variables. So: $f_X(x) = \frac{1}{\sqrt{2\pi}}e^{\frac{-1}{2}x^2}$ and $f_Y(y) = \frac{1}{\sqrt{2\pi}}e^{\frac{-1}{2}y^2}$. Now I am interested in the probability density function (PDF) $f_Z(z)$ when $Z=\frac{1}{2}(X^2+Y^2)$. 
I know some things about the sum of two normally distributed random variables, for example: $Z_1 = X +Y$ gives that $Z_1 \sim N(0,2)$ by the use of convolution.
How to obtain (efficiently) the PDF for $Z$, again by the use of convolution?
 [/INST] $$\mathbb{P}[X^2+Y^2\leq R^2]=\frac{1}{2\pi}\iint_{x^2+y^2\leq r^2}e^{-\frac{x^2+y^2}{2}}\,dx\,dy=\frac{1}{2\pi}\int_{0}^{2\pi}\int_{0}^{R}\rho e^{-\rho^2/2}\,d\rho\,d\theta$$
so:
$$\mathbb{P}[X^2+Y^2\leq R^2]=1-e^{-R^2/2}$$
and $X^2+Y^2$ has an exponential distribution with parameter $\lambda=\frac{1}{2}$.
</s>