# Deriving exponential distribution from sum of two squared normal random variables

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

• Hint: convert the integral to polar coordinates, where $Z$ becomes the square of the radius. – whuber Feb 11 '17 at 23:27

First use the joint pdf of $X$ and $Y$ and switch to polar coordinates, then $$\mathbb{P}(Z\leq z)=\mathbb{P}(X^2+Y^2\leq z)=\frac{1}{\pi}\int_{x^2+y^2\leq z}e^{-x^2+y^2}\;dxdy=\frac{1}{\pi}\int_{0}^{2\pi}\int_0^{\sqrt{z}}e^{-r^2}r\;drd\theta$$ $$=2\int_0^{\sqrt{z}}re^{-r^2}\;dr$$ Now if we set $u=r^2$ then we get $$\mathbb{P}(Z\leq z)=\int_0^ze^{-u}\;du$$
so $Z$ is exponentially distributed with rate parameter $\lambda = 1$.
• How do we know the joint CDF of $X^2$ and $Y^2$? – gwg May 16 at 21:55