let $X$ be a Gaussian random vector in $R^n$ such that
$$X \sim \mathcal{N}(\mathbf{0}, \mathbf{I_n}),$$
How can I find the PDF of $\frac{X}{\|X\|}$?
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By rotation invariance / symmetry, the PDF is the uniform distribution on the unit sphere, so the PDF is the constant value $\frac{1}{S_{n-1}}$ where $S_{n-1} = \frac{n \pi^{n/2}}{\Gamma(\frac{n}{2}+1)}$ is the surface area of the unit sphere in $\mathbb{R}^n$.
angryavian
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sorry , Indeed I know this result ,however I can't prove that .and I see some answer on this website but not quite get it . – ShaoyuPei Jan 11 '19 at 01:22
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What can be said when $X\sim\mathcal{N}(\mu,\Sigma)$? – nullgeppetto Feb 20 '19 at 19:43
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@ShaoyuPei https://math.stackexchange.com/questions/3120506/on-the-distribution-of-a-normalized-gaussian-vector – angryavian Feb 21 '19 at 20:34
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This doesn't seem to integrate to 1. – Phoenix May 01 '23 at 20:24