A New Geometric Probability Technique for an N-dimensional Sphere and Its Applications to Physics

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.