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

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. 1. Introduction The Poisson and Binomial distributions are a good approximation for many random phenomena in areas such as telecommunications and reliability engineering, as well as the biological and managerial sciences [1, 2]. Let be a Poisson distributed random variable having mean , and let represent the cumulative distribution function (CDF) of with nonnegative integer support : Similarly, let be a Binomially distributed random variable with parameters and , and let represent the CDF of for integer support : Also, let the th quantiles of and for be obtained from the functions and : Due to numerical and complexity issues, evaluation of the exponential and Binomial summations in (1) and (2) through recursive operations is only practical for small values of the input parameters ( or and ). Instead, a better solution is to evaluate the CDFs directly through either their incomplete Beta/Gamma function representations which can be approximated to high precision by continued fractions or asymptotic expansions [3]. With respect to the quantiles of the distributions given by (3) and (4), no methods to exactly evaluate these functions without iterating the exponential/Binomial sums—or alternately employing a search until the required conditions are satisfied—seem to be known. Typically, a binary search to determine the smallest satisfying (3) or (4) evaluating the respective CDF at each step would be a better general solution, given some initial upper bound for . Such methods (and related variants) are now employed very effectively in modern commercial and research-based statistical packages. In some situations, one may desire simpler solutions to