Deriving exponential distribution from sum of two squared normal random variables

Question

Let $X$, $Y$ be i.i.d. random varibales with distribuition $\mathcal{N}(0,1)$ and $Z = X^2 + Y^2$. I'd like to prove based on $X$ and $Y$ pdf's that $Z$ has exponential distribuition.

Answer 1

First use the joint pdf of $X$ and $Y$ and switch to polar coordinates: if $z>0$ then $$ \mathbb{P}(Z\leq z)=\mathbb{P}(X^2+Y^2\leq z)=\frac{1}{2\pi}\int_{x^2+y^2\leq z}e^{-\frac{x^2+y^2}{2}}\;dxdy=\frac{1}{2\pi}\int_{0}^{2\pi}\int_0^{\sqrt{z}}e^{-\frac{r^2}{2}}r\;drd\theta$$ $$=\int_0^{\sqrt{z}}re^{-\frac{r^2}{2}}\;dr $$ Now if we set $u=r^2$ then we get $$ \mathbb{P}(Z\leq z)=\frac{1}{2}\int_0^ze^{-\frac{u}{2}}\;du$$

so $Z$ is exponentially distributed with parameter $\frac{1}{2}$.