Approximating the Number of Successes in Independent Trials: Binomial versus Poisson

• Published in: The Annals of applied probability 2002-11, Vol.12 (4), p.1139-1148
• Main Authors: Choi, K. P., Xia, Aihua
• Format: Article
• Language: English
• Subjects: 60E15; 60F05; Approximation; Binomial distribution; Binomial distributions; Binomials; Integers; Mathematical inequalities; Poisson distribution; Random variables; Statistical variance; total variation metric
• ISSN: 1050-5164; 2168-8737
• DOI: 10.1214/aoap/1037125856

Let I1,I2,... ,In be independent Bernoulli random variables with P(Ii=1)=1- P(Ii=0)=pi, 1≤ i≤ n, and W=∑i=1nIi,λ = EW=∑i=1npi. It is well known that if pi'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=\lfloor \lambda \rfloor $, the greatest integer ≤ λ, and set δ = λ - $\lfloor \lambda \rfloor $, and κ be the least integer more than or equal to max {λ 2/(r-1-(1+δ)2),n}. In this paper, we prove that, if $r>1+(1+\delta)^{2}$, then $d_{\kappa}