Probabilities for the Size of Largest Clusters and Smallest Intervals

Abstract

Abstract Given N points distributed at random on [0,1), let np be the size of the largest number of points clustered within an interval of length p. Previous work finds Pr (np≥n), for n>N/2, and for n≤N/2, p=1/L, L an integer. The formula for the case p=1/L is in terms of the sum of L×L determinants and is not computationally feasible for large L. The present paper derives such a computational formula.