Say I have two random variables, $x,y$ that are independent and normally distributed, $\mathcal{N}(0,1)$. What is the probability distribution function of $r$ and $r^2$ where $r=\sqrt{x^n+y^n}$ and $n$ is a positive even integer? It is known that for $n=2$ we get a Rayleigh distribution for $r$, but can something be said for higher values of $n$?
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