On beta type-1 distributed random points in an n-ball

This article considers the problem of finding the exact density of the r-content of the simplicial convex hull of r+1 independent points in Rⁿ ” Consider r+1 independent and identically distributed points in a unit n–ball such that p of them are in the interior and r+l?p of them are on the surface of the unit n-ball., Consider the case when each point is type-1 beta distributed,, These points determine almost surely via their convex hull a unique r-simplex in Rⁿ Ihe problem of getting the exact density of the r-content of this random r-simplex is transformed into a problem in multivariate statistical analysis connected with the distribution theory of test statistics., Thus various representations for the exact density are given in this article.