# PDF of normalized gaussian random vector

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

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 $$\mathbb{R}^n$$.
• What can be said when $X\sim\mathcal{N}(\mu,\Sigma)$? – nullgeppetto Feb 20 at 19:43