Why is $S = X^2 + Y^2$ distributed $Exponential(\frac{1}{2})$? Let $X$ and $Y$ be independent, standard normally distributed random variables ($\sim Normal(0, 1)$). Why is $S = X^2 + Y^2$ distributed $Exponential(\frac{1}{2})$?
I understand that an exponential random variable describes the time until a next event given a rate at which events occur. However, I do not understand what this has to do with $S = X^2 + Y^2$.

Sum of 2 standard normals is chi squared with 2 degrees of freedom.
A chi squared with 2 degrees of freedom is equivalent to a exponential with parameter =$1/2.$
