If $(x_1,x_2,...,x_N)$ is a uniformly randomly chosen point on a hypersphere of a dimension $N$ with the radius $R$ (center in origin). What is the probability distribution of any coordinate?
Done so far (2-D): From coordinates in polar form, and angle chosen uniformly I've calculated the distribution function, which is equal to $f(x) = \frac{1}{\pi \sqrt{r^2-x^2}}$ (here the radius is denoted with $r$, and coordinate with $x$). Cumulative distribution function is $F(x)=1-\frac{cos^{-1}\left(\frac{x}{r}\right)}{\pi}\text{ ,}$ if it should be of any use.
I've tried to calcualte probability distribution of coordinate in 3-D by denoting $f(x)$ given above as $f(x|r)$, and knowing that (calculated similarly as the distribution above) $f(r) = \frac{1}{\pi \sqrt{R^2-(R-r)^2}}\text{ ,}$I've tried to calculate the distribution $f(x)$ in 3-D by integrating $f(x,r)=f(x|r) f(r)$ from $0$ to $R$, but Wolfram Alpha timed out (link). Mathematica returned me the inputed integral for
Integrate[(2/Pi^2)*(1/Sqrt[(r^2 - x^2)*(R^2 - (R - r)^2)]), {r, 0, R}]