Distribution of number of observations required to obtain a cover for the support of a uniform distribution

Abstract

AbstractFor a given positive number ‘δ′, we consider a sequence of δ− neighborhoods of the independent and identically distributed (i.i.d.) random variables, from a U(0,1) distribution, and “stop as soon as their union contains the interval (0,1).” We call such a union “a cover.” To find the distributions of N(δ), the stopping time random variable, we need the joint distribution of order statistics from a U(0,1) distribution. For each δ>0 and n=1,2,…, we obtain a general expression for P(N(δ)≤n), and for a fixed value of δ, it is the distribution function of N(δ). For a given n, let Δ(n) be the minimum value of δ, so that the union of the n δ− neighborhoods of the first n observations contains the interval (0,1). Because N(δ)≤n if and only if Δ(n)≤δ, the distributions of Δ(n) can be obtained by fixing n in the general expression for P(N(δ)≤n). To describe the impact of δ on the distribution of N(δ) and that of n on Δ(n), we sketch the graphs of distribution functions and the empirical distribution functions.Keywords: Neighborhoodstopping ruleuniform distributionSubject Classification: MSC 2020: 26B1562E15 ACKNOWLEDGMENTThanks to S. B. Patil and S. R. Rattihalli for their computational assistance.DISCLOSUREThe author has no conflicts of interest to report.