Approximating the number of successes in independent trials: Binomial versus poisson

Abstract

Let I1, I2,..., In be independent Bernoulli random variables with ℙ(Ii = 1) = 1 - ℙ(I i = 0) = pi, 1 ≤ i ≤ n, and W = ∑ i=1n Ii, λ = double struct E sign W = ∑i=1n pi. It is well known that if p i's are the same, then W follows a binomial distribution and if pi's are small, then the distribution of W, denoted by ℒW, can be well approximated by the Poisson(λ). Define r = ⌊λ⌋, the greatest integer ≤ λ, and set δ = λ - ⌊λ⌋, and κ be the least integer more than or equal to max{λ2/(r - 1 - (1 + δ)2),n}. In this paper, we prove that, if r > 1 + (1 + δ)2, then d κ < dκ+1 < dκ+2 < ⋯< dTV(ℒW, Poisson(λ)), where dTV denotes the total variation metric and dm = dTV(ℒW, Bi(m, λ/m)), m ≥ κ. Hence, in mode..