Open Access
Translator Disclaimer
November 2002 Approximating the number of successes in independent trials: Binomial versus Poisson
K. P. Choi, Aihua Xia
Ann. Appl. Probab. 12(4): 1139-1148 (November 2002). DOI: 10.1214/aoap/1037125856

Abstract

Let $I_1,I_2,\ldots,I_n$ be independent Bernoulli random variables with $\mathbb{P}(I_i=1)=1-\mathbb{P}(I_i=0) =p_i$, $1\le i\le n$, and $W=\sum_{i=1}^nI_i$, $\lambda=\mathbb{E}W=\sum_{i=1}^np_i$. It~is well known that if~$p_i$'s are the same, then~$W$ follows a~binomial distribution and if~$p_i$'s are small, then the distribution of~$W$, denoted by~$\mathcal{L} W$, can be well approximated by the $\mathop{\mathrm{Poisson}}(\lambda)$. Define $r=\lfloor\lambda\rfloor$, the greatest integer~$\le\lambda$, and set $\delta=\lambda-\lfloor \lambda \rfloor$, and~$\kappa$ 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 \[ d_\kappa<d_{\kappa+1}<d_{\kappa+2}<\cdots<d_{\mathit{TV}} (\mathcal{L} W,\mbox{Poisson}(\lambda)), \] where $d_{\mathit{TV}}$ denotes the total variation metric and $d_m=d_{\mathit{TV}}(\mathcal{L} W,\break\Bi(m,\lambda/m))$, $m\ge\kappa$. Hence, in modelling the distribution of the sum of Bernoulli trials, Binomial approximation is generally better than Poisson approximation.

Citation

Download Citation

K. P. Choi. Aihua Xia. "Approximating the number of successes in independent trials: Binomial versus Poisson." Ann. Appl. Probab. 12 (4) 1139 - 1148, November 2002. https://doi.org/10.1214/aoap/1037125856

Information

Published: November 2002
First available in Project Euclid: 12 November 2002

zbMATH: 1019.60018
MathSciNet: MR1936586
Digital Object Identifier: 10.1214/aoap/1037125856

Subjects:
Primary: 60F05
Secondary: 60E15

Keywords: Binomial distribution , Poisson distribution , total variation metric

Rights: Copyright © 2002 Institute of Mathematical Statistics

JOURNAL ARTICLE
10 PAGES


SHARE
Vol.12 • No. 4 • November 2002
Back to Top