Approximating the number of successes in independent trials: Binomial versus Poisson
成果类型：
Article
署名作者：
Choi, KP; Xia, AH
署名单位：
National University of Singapore; University of Melbourne
刊物名称：
ANNALS OF APPLIED PROBABILITY
ISSN/ISSBN：
1050-5164
发表日期：
2002
页码：
1139-1148
关键词：
摘要：
Let I-1, I-2, ..., I-n be independent Bernoulli random variables with P(I-i = 1) = 1 - P(I-i = 0) = p(i), 1 less than or equal to i less than or equal to n, and W = Sigma(i)(n) = (1) I-i, lambda = EW = Sigma(i)(n) = p(i). 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 X W, can be well approximated by the Poisson(lambda). Define r = [lambda], the greatest integer less than or equal to lambda and set delta = lambda - [lambda] and k be the least integer more than or equal to max{lambda(2)/(r - 1 - (1 + delta)(2)),n}. In this paper, we prove that, if r > 1 + (1 + delta)(2), then d(k) < d(k+1) < d(k+2) <... < d(TV) (LW, Poisson(lambda)), where dTV denotes the total variation metric and d(m) = d(TV) (LW, Bi(m, lambda/m)), m greater than or equal to k. Hence, in modelling the distribution of the sum of Bernoulli trials, Binomial approximation is generally better than Poisson approximation.