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Let $X \sim \mathcal{N}(\mu, \Sigma) \in \mathbb{R}^d$ follow a multivariate normal distribution. Then what's the distribution (PDF, CDF etc.) of $X?$ When $\mu = 0, \Sigma = I_d,$ we know that $||X||\sim \chi(d),$ the chi distribution, but I don't see why that'd directly tell me anything about the distribution of $||X||$ in general cases of $\mu, \Sigma?$

If needed, you can assume that $\mu=0$ or $\Sigma$ is diagonal (not identity) or a combination of both. For the moment, I'm more particularly interested in $\mathbb{E}||X||, var[||X||]$. References very much appreciated as well!

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    $\begingroup$ en.wikipedia.org/wiki/Generalized_chi-squared_distribution me thinks $\endgroup$
    – Brendan McKay
    Commented Apr 28, 2020 at 12:20
  • $\begingroup$ @BrendanMcKay Thank you, but I think it gives us the distribution of $||X||^2,$ but not of $||X||,$ so it seems we need to use PDF transformation for $W:=\sqrt{V}$ to obtain the PDF for $||X||...$ $\endgroup$
    – Learning math
    Commented Apr 28, 2020 at 13:22

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