On the Convergence of Binomial to Poisson Distributions

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.