How to poison Poisson (when approximating binomial tails)*

Abstract

Summary In these days some sportsmen are given dangerous drugs in order to stimulate them into still higher efforts. The present paper wants to add some poison to the classical approximation of a binomial by a Poisson distribution. The relatively harmless drug of a little extra calculation shall be seen to result in a much better accuracy. This paper reports both on theoretical considerations, following from series expansions of Poisson‐type parameters, and on extensive numerical investigations of accuracy by means of an Electrologica X8 computer. The main conclusion is that the probability of at most k successes in n Bernoulli trials with success probability p <.5 can be very closely approximated by the probability of at most k events in a Poisson distribution with expectation not np but (12–2 p ) n‐ 7 k ζ= (12–8 p )n‐k+k/n np. In the case p > 5 one should make p < .5 by a reversal, i.e. the interchanging of successes and failures, before the application of a Poisson‐type approximation (with one exception mentioned in section 4). For the probability of at least j successes one should approximate the complementary probability of at most j—1 successes. After an introductory section 1 and a discussion of measures of accuracy in section 2, the existing Poisson‐type approximations are treated in section 3. The fourth section introduces a new parameter and discusses the advantages of reversal in other situations than p > 5. In section 5 a series expansion for the exact Poisson parameter is derived. Two new parameters, among which is ζ, are introduced in section 6, where the series expansions for parameters given in table 4 lead to conclusions about the various approximations, summarized in table 6. The final section 7 gives some numerical results, which confirm to a large extent the conclusions from the asymptotic expansions.