Let $X \in \mathbb{R}^n$ be random variable following a multivariate Gaussian distribution $\mathcal{N}(\mu, \Sigma)$. Do we know the distribution for its norm $r = ||X||_2 = \sqrt{X^\top X}$?
I understand that $X^\top X$ follows a generalized chi-squared distribution, but I can't find the distribution for $\sqrt{X^\top X}$. I would appreciate your help.