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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 $\mathbb{R}^n$.

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  • $\begingroup$ 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 . $\endgroup$ – ShaoyuPei Jan 11 at 1:22
  • $\begingroup$ What can be said when $X\sim\mathcal{N}(\mu,\Sigma)$? $\endgroup$ – nullgeppetto Feb 20 at 19:43
  • $\begingroup$ @ShaoyuPei math.stackexchange.com/questions/3120506/… $\endgroup$ – angryavian Feb 21 at 20:34

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