The broken stick model. The optimality property of the uniform distribution

Abstract

The problem. Let I = [0, l] be a stick of length l. Let (Xn) 1≤n≤N+1 be iid absolutely continuous random variables valued into I and let F be their distribution. Let (Xn:N ) 1≤n≤N+1 be their order statistic and Yn = Xn:N − X(n−1):N , 1≤n≤N+1 be the corresponding spacings. Here X0:N = 0.