Finding The Distribution Of A Random Variable From Its Moment Function

ABSTRACT

Consider the problem of r t / randomly distributed points in a unit n-ball and the convex hull created by these points. Let ~ 11 be r.t times the r-content of an rsimplex whose p vertices are in the interior and r + /- p vertices on the boundary of a unit n-ball. Explicit expressions for the exact distribution functions of ~,, are given when r 1 / points are independently, and identically distributed according to the Uniform distribution. The exact distributions are obtained using the technique · of Inverse Mellin transforms with the help of the moment functions. The technique is illustrated for the general case p =- r -1 1 and a particular case p =3, r -2 . Various representations of the distributions in psi and the generalized zeta functions are given. These representations are also given in the most general case as an H-function distribution.