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Title: Approximating the number of successes in independent trials: Binomial versus poisson
Authors: Choi, K.P. 
Xia, A.
Keywords: Binomial distribution
Poisson distribution
Total variation metric
Issue Date: Nov-2002
Citation: Choi, K.P., Xia, A. (2002-11). Approximating the number of successes in independent trials: Binomial versus poisson. Annals of Applied Probability 12 (4) : 1139-1148. ScholarBank@NUS Repository.
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=1 n Ii, λ = double struct E sign W = ∑i=1 n 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 modelling the distribution of the sum of Bernoulli trials, Binomial approximation is generally better than Poisson approximation.
Source Title: Annals of Applied Probability
ISSN: 10505164
DOI: 10.1214/aoap/1037125856
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