## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 42, Number 5 (1971), 1735-1736.

### On the Convergence of Binomial to Poisson Distributions

Gordon Simons and N. L. Johnson

#### Abstract

Following the work of Poisson (1837), there has been considerable practical and theoretical interest in how well the Poisson distribution approximates the binomial distribution. The approximation, which was initially suggested by a limit theorem (see (1) below), has been shown in numerical examples to be very good for certain binomial parameters within a useful range. (See Feller (1950), page 143.) Subsequently, (nonasymptotic) theoretical results have confirmed the approximation's accuracy. (See (2.1) and (2.2) below.) The purpose of this note is to demonstrate, with an elementary argument, that the binomial distributions converge very strongly to the Poisson distributions.

#### Article information

**Source**

Ann. Math. Statist., Volume 42, Number 5 (1971), 1735-1736.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177693172

**Digital Object Identifier**

doi:10.1214/aoms/1177693172

**Mathematical Reviews number (MathSciNet)**

MR345171

**Zentralblatt MATH identifier**

0235.60033

**JSTOR**

links.jstor.org

#### Citation

Simons, Gordon; Johnson, N. L. On the Convergence of Binomial to Poisson Distributions. Ann. Math. Statist. 42 (1971), no. 5, 1735--1736. doi:10.1214/aoms/1177693172. https://projecteuclid.org/euclid.aoms/1177693172