A set of points are uniformly i.i.d in the unit n-hypersphere centered at the origin, what is the probability that a point has the radius $r$, i.e what is the probability distribution of $r\in[0,1]$.
For example in 2D this is proportional to the circumference of $r$ i.e $2\pi r$ and in a sphere this is the surface area.
I've read this answer What's the expected radius of a hypersphere but i'm not particularly clear on the distribution they give.