Random division of an interval

Abstract

We consider n − 1 points x j ( j = 1, …, n − 1) selected independently at random from the interval (0, 1), the distribution of any x j being the rectangular distribution The points x j divide the interval (0,1) into n intervals. Let λ be the sum of the lengths of the k largest of these n intervals. It is the purpose of this paper to give the exact sampling distribution of λ for any given n and k (see equation (5)). Fisher ( 1 ) in 1929 solved the problem for k = 1. We first investigate the distribution of μ = 1 − λ.