  A new formalism is presented for analytically obtaining the probability
density function, \( P_{n}(s) \), for the distance between two random points in
an \( n \)-dimensional sphere of radius \( R \). Our formalism allows \(
P_{n}(s) \) to be calculated for a sphere having an arbitrary density
distribution, and reproduces the well-known results for the case of a sphere
with uniform density. The results find applications in stochastic geometry,
probability distribution theory, astrophysics, nuclear physics, and elementary
particle physics.
