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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  • ANDERSON, T. W. and SAMUELS, S. (1967). Some inequalities among binomial and Poisson probabilities. Proc. Fifth Berkeley Sy mp. Math. Statist. Probab. 1 1-12. Univ. California Press, Berkeley.
  • BARBOUR, A. D. and HALL, P. (1984). On the rate of Poisson convergence. Math. Proc. Cambridge Philos. Soc. 95 473-480.
  • BARBOUR, A. D., HOLST, L. and JANSON, S. (1992). Poisson Approximation. Oxford Univ. Press, Oxford.
  • BLISS, C. and FISHER, R. A. (1953). Fitting the negative binomial distribution to biological data. Biometrics 9 174-200.
  • BROWN, T. C. and XIA, A. (2001). Stein's method and birth-death processes. Ann. Probab. 29 1373-1403.
  • CHEN, L. H. Y. (1975). Poisson approximation for dependent trials. Ann. Probab. 3 534-545.
  • DEHEUVELS, P. and PFEIFER, D. (1986). A semigroup approach to Poisson approximation. Ann. Probab. 14 663-676.
  • EHM, W. (1991). Binomial approximation to the Poisson binomial distribution. Statist. Probab. Lett. 11 7-16.
  • HAND, D. J., DALY, F., MCCONWAY, K. J. and OSTROWSKI, E. (1996). A Handbook of Small Data Sets. Chapman and Hall, London.
  • HOEFFDING, W. (1956). On the distribution of the number of successes in independent trials. Ann. Math. Statist. 27 713-721.
  • POISSON, S. D. (1837). Recherches sur la probabilité des jugements en matière criminelle et en matière civile, précedées des règles générales du calcul des probabilités. Bachelier, Paris.
  • ROOS, B. (2000). Binomial approximation to the Poisson binomial distribution: The Krawtchouk expansion. Theory Probab. Appl. 45 258-272.
  • SAMUELS, S. M. (1965). On the number of successes in independent trials. Ann. Math. Statist. 36 1272-1278.
  • SOON, S. Y. T. (1996). Binomial approximation for dependent indicators. Statist. Sinica 6 703-714.
  • STUDENT (1907). On the error of counting with a haemacy tometer. Biometrika 5 351-360.
  • VOLKOVA, A. YU. (1995). A refinement of the central limit theorem for sums of independent random indicators. Theory Probab. Appl. 40 791-794.
  • VON BORTKEWITSCH, L. (1898). Das Gesetz der kleinen Zahlen. Teubner, Leipzig.