TITLE: Marginal Distribution of Uniform Vector on Sphere
QUESTION [5 upvotes]: Suppose that $U=(U_1,\ldots,U_n)$ has the uniform distribution on the unit sphere $S_{n-1}=\{x\in\mathbb R^n:\|x\|_2=1\}$.
I'm trying to understand the marginal distributions of individual components of the vector $U$, say, without loss of generality, $U_1$.
So far, this is what I have: I know that $U$ is equal in distribution to $Z/\|Z\|_2$, where $Z$ is an $n$-dimensional standard gaussian vector. Thus, $U_1$ is equal in distribution to $Z_1/\|Z\|_2$. Then, we could in principle compute the distribution of $U_1$ as
$$P[U_1\leq t]=P\big[Z_1\leq\|Z\|_2t\big],$$
but the right-hand side above using conventional means is horribly messy (i.e., nested integrals over the set $[x_1\leq\|x\|t]$).
Is there a more practical/intuitive way of computing this distribution?

REPLY [4 votes]: I hope that this is still relevant to you. 
The answer can be found in eq. 1.26: 
Fang, Bi-Qi; Fang, Kai-Tai, Symmetric multivariate and related distributions,  (2017). (https://books.google.co.il/books?hl=iw&lr=&id=NL1HDwAAQBAJ&oi=fnd&pg=PT10&ots=u5cnMFtVxP&sig=ZgnWbGkG8qdVARoJ64mfm9fcag0&redir_esc=y#v=onepage&q&f=false).
The marginal density of $(z_1/\Vert \textbf{z} \Vert, ..., z_k /\Vert \textbf{z} \Vert )$ is 
$$\frac{\Gamma(n/2)}{\Gamma((n-k)/2) \pi^{k/2}} \left(1 - \sum_{i=1}^k z_i^2 \right)^{(n-k)/2 -1}$$ 
This can be derived from the Dirichlet distribution.