Distribution of subvector of vector uniform on a sphere

Question

For a random vector $\mathbf{X} = [x_1,x_2,...,x_n] \in \mathbb{R}^n$ uniformly distributed on the surface of the unit sphere, the PDF is the inverse of the surface $$f_\mathbf{X}(\mathbf{x}) = 2\pi^{-n/2}\Gamma(n/2) \delta(\left\lVert \mathbf{x}\right\rVert = 1) \tag{1}$$ where $\delta$ is the delta function.

I am interested in $$f_{X_1,...,X_{n-1}}(x_1,...,x_{n-1}) = \int_{-1}^1 f_\mathbf{X}(\mathbf{x}) dx_n \\= 2\pi^{-n/2}\Gamma(n/2) \int_{-1}^1 \delta(x_n^2 = 1 - x_1^2-...-x_{n-1}^2) dx_n = 2\pi^{-n/2}\Gamma(n/2) \times 2$$ because there is 2 points $x_n^2 = \pm\sqrt{1 - x_1^2-...-x_{n-1}^2}$.

I am not sure this is correct because if I continue like that, the marginal distribution $f_{X_1}(x_1)$ will be something does not depend on $x_1$ which is not true.

Thanks for hints.

Answer 1

One can think about this problem very geometrically: Let $F(r)$ be defined as the surface area of the boundary of a $n$ dimensional ball of radius $r$. Then the pdf of $X_1$ is $f_1(x)= \frac{F(r)}{\int_{-1}^{1} F(r) dx}$ where $r=\sqrt{1-x^2}$.

Answer 2

[1] V. I. Khokhlov, "The Uniform Distribution on a Sphere in ${\bf R}^S$. Properties of Projections. I.", Theory Probab. Appl., 50(3), 386–399, 2006.

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 unit sphere $S_n = \{(x_1, x_2, \cdots, x_n) \in \mathbb{R}^n : \ x_1^2 + x_2^2 + \cdots + x_n^2 = 1\}$.

Then, $X_1$ has the probability density $$p_{X_1}(x_1) = \frac{\Gamma(\frac{n}{2})}{\sqrt{\pi}\Gamma(\frac{n-1}{2})}(1-x_1^2)^{\frac{n-3}{2}}, \ \ - 1 < x_1 < 1.$$ If $n=3$, it is $f_{X_1}(x_1) = \frac{1}{2}, \ -1 < x_1 < 1$ which is a uniform distribution.

$(X_1, X_2, \cdots, X_k)$ ($k \le n-2$) has the joint probability density \begin{align} f_{X_1, X_2, \cdots, X_k}(x_1,x_2, \cdots, x_k) &= \frac{\Gamma(\frac{n}{2})}{\pi^{k/2}\Gamma(\frac{n-k}{2})} (1 - 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 < 1. \end{align}