Density of the first $k$ coordinates of a uniform random variable

Question

Suppose that $X$ is distributed uniformly in the $n$-sphere $\sqrt{n}\mathbf{S}^{n-1} \subset \mathbf{R}^n$. Then apparently the distribution of $(X_1, \dots, X_k)$, the first $k < n$ coordinates of $X$ has density $p(x_1, \dots, x_k)$ with respect to Lebesgue measure in $\mathbf{R}^k$, moreover if $r^2 = x_1^2 + \cdots + x_k^2$, then it is proportional to $$ \left(1 - \frac{r^2}{n}\right)^{(n-k)/2 - 1}, \quad \text{if}~0 \leq r^2 \leq n, $$ and otherwise is 0. I tried to compute this using the fact that $(X_1, \dots, X_k) \stackrel{\rm d}{=} \sqrt{n} (g_1, \dots, g_k)/\sqrt{g_1^2 + \cdots + g_n^2}$, when $g_i$ are iid standard normal variables, but was unable to simplify the integrals. Does anyone know/can point me to a place where this density is derived?

Answer 1

[1] A. J. Stam, "Limit Theorems for Uniform Distributions on Spheres in High-Dimensional Euclidean Spaces", Journal of Applied Probability, Vol. 19, No. 1 (Mar., 1982), pp. 221-228

Let $n\ge 3$ be an integer. Let $(X_1, X_2, \cdots, X_n)$ be a random vector with a uniform distribution on the sphere $S_n(\sqrt{n}) = \{(x_1, x_2, \cdots, x_n) \in \mathbb{R}^n : \ x_1^2 + x_2^2 + \cdots + x_n^2 = n\}$.

Let $1\le k\le n-2$ be an integer. $(X_1, X_2, \cdots, X_k)$ has the joint probability density \begin{align} f_{X_1, X_2, \cdots, X_k}(x_1,x_2, \cdots, x_k) &= n^{-n/2 + 1}\frac{\Gamma(\frac{n}{2})}{\pi^{k/2}\Gamma(\frac{n-k}{2})} (n - x_1^2 - x_2^2 - \cdots - x_k^2)^{\frac{n-k-2}{2}}, \\ &\quad (x_1, x_2, \cdots, x_k)\in \mathbb{R}^k, \ 0 < x_1^2 + x_2^2 + \cdots + x_k^2 < n. \end{align}