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

Abstract

For 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.