The Annals of Applied Probability

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

K. P. Choi and Aihua Xia

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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.

Article information

Ann. Appl. Probab., Volume 12, Number 4 (2002), 1139-1148.

First available in Project Euclid: 12 November 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60E15: Inequalities; stochastic orderings

Binomial distribution Poisson distribution total variation metric


Choi, K. P.; Xia, Aihua. Approximating the number of successes in independent trials: Binomial versus Poisson. Ann. Appl. Probab. 12 (2002), no. 4, 1139--1148. doi:10.1214/aoap/1037125856.

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