Suppose $\mathbf{x}$ is uniformly sampled from the surface of n-sphere. What is the distribution of $x_1^2$?
This answer shows that $x_1$ is distributed as shifted Beta so I can plug it into Mathematica to get distribution of $x_1^2$ for some specific $n$, wondering if there's an elegant way to derive the formulas below:
$$\begin{matrix} n=1&\frac{1}{\pi \sqrt{((1-x) x)}}\\ n=2& \frac{1}{2 \sqrt{x}}\\ n=3& \ldots\\ n=4&\frac{3 (1-x)}{4 \sqrt{x}}\\ n=5&\ldots\\ n=6&\frac{15 (x-1)^2}{16 \sqrt{x}}\\ n=7&\ldots\\ n=8&\frac{35 (1-x)^3}{32 \sqrt{x}} \end{matrix}$$
