On uniformly distributed random points in an n-Ball

Abstract

Consider a set of r+1 independently and identically and uniformly distributed random points X₀, X₁,…,X_r in RⁿThese points determine almost surely via their convex hull a unique r-simplex in R^e This article deals with the exact density of the r-content of this random r-simplex when the points are such that p of them are in the interior and r+l−p of them are on the surface of a unit n-ball. This problem is transformed into a distribution problem connected with multivariate test statistics. Various possible representations of the exact density in the general case, are also pointed out.

Keywords:

• Uniformly distributed points in an n-ball
• random r-contents
• exact density of r-contents
• different representations for the exact density