Distance between random points in a cube

Abstract

The explicit forms of the exact density and the exact arbitrary moments of the distance between two random points when the points are independently and uniformly distributed inside a cube are given. Special cases considered include the density and arbitrary moments of the distance between two independent random points where one is inside the cube and the other is on a face, points on adjacent faces, points on opposite faces and random points inside a square. Exact arbitrary moments are obtained in terms of Gauss'hypergeometric functions and Appell's functions or Lauricella functions.