There is [a simple algorithm to pick a random point ON an $n$-dimensional hypersphere](https://mathoverflow.net/questions/136314/what-is-a-good-method-to-find-random-points-on-the-n-sphere-when-n-is-large).


Is there one to sample a point from inside it? (Sampling points from a hypercube and rejecting them if they are not inside the hypersphere would scale extremely poorly with $n$, as the vast majority of points would be rejected)



Sampling a uniformly distributed point INSIDE a hypersphere?
Choose a uniform point $X$ on the unit hypersphere, then multiply it by $U^{1/n}$ where $U \sim U(0,1)$ is independent of $X$.


