Some Suggestions for Teaching About Normal Approximations to Poisson and Binomial Distribution Functions

Abstract

In this article we review two historical approximations to the Poisson and binomial cumulative distribution functions (CDFs); that is, the Wilson–Hilferty and Camp–Paulson approximations. Both of these approximations reduce to standard normal formulas that produce very accurate estimates of the Poisson and binomial CDFs, and are thus quite simple to implement. Additionally, in an upper-division undergraduate or master’s level probability and inference course, the derivation of these approximations presents a nice opportunity to introduce and study the distributional relationships between the gamma and Poisson CDFs, and the binomial, beta, and F CDFs. This article presents the basic theorems and lemmas needed to derive each approximation, along with some relevant examples that compare and contrast the precision of these approximations with their large-sample, limiting normal counterparts.