Table of Contents
ISRN Probability and Statistics
Volume 2013, Article ID 412958, 6 pages
http://dx.doi.org/10.1155/2013/412958
Research Article

Improved Inequalities for the Poisson and Binomial Distribution and Upper Tail Quantile Functions

Electronics & Control Group, Teesside University, Middlesbrough TS1 3BA, UK

Received 29 October 2013; Accepted 25 November 2013

Academic Editors: V. Makis and A. Volodin

Copyright © 2013 Michael Short. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The exact evaluation of the Poisson and Binomial cumulative distribution and inverse (quantile) functions may be too challenging or unnecessary for some applications, and simpler solutions (typically obtained by applying Normal approximations or exponential inequalities) may be desired in some situations. Although Normal distribution approximations are easy to apply and potentially very accurate, error signs are typically unknown; error signs are typically known for exponential inequalities at the expense of some pessimism. In this paper, recent work describing universal inequalities relating the Normal and Binomial distribution functions is extended to cover the Poisson distribution function; new quantile function inequalities are then obtained for both distributions. Exponential bounds—which improve upon the Chernoff-Hoeffding inequalities by a factor of at least two—are also obtained for both distributions.