PDF of normalized gaussian random vector

Question

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 $\frac{X}{\|X\|}$?

Answer 1

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 $\mathbb{R}^n$.