I've been wondering if there is a result on normalized normal vectors. That is, if $X \sim N_d(\mu,\Sigma)$, where $N_d(\mu,\Sigma)$ is a $d$-dimensional multivariate normal distribution with mean vector $\mu \in \mathbb{R}^d$ and symmetric positive definite covariance matrix $\Sigma \in \mathbb{R}^{d\times d}$, I would like to know the distribution of $W = X/\sqrt{X^\prime X}$.
This is a type of norm-restricted normal distribution, i.e., $X \sim N_d(\mu,\Sigma)I(\|X\| = 1)$, but I wonder if there are more mathematical properties available for this kind of distribution, e.g., moment-generating function, characteristic function, CDF, or PDF.
