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.
Highlights
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]
A binary search to determine the smallest k satisfying (3) or (4) evaluating the respective cumulative distribution function (CDF) at each step would be a better general solution, given some initial upper bound for ISRN Probability and Statistics k
It was not explicitly denoted as such in [5], it is easy to see that D(p, c) represents the Kullback-Leibler ( KL ) divergence between two Bernoulli variables with respective probabilities of success p and c; nD(p, c) represents the KL divergence of n summed pairs of such variables. This observation allows the relatively straightforward extension of the above result to the case of a Poisson distributed random variable, which is given in Theorem 2
Summary
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]. A binary search to determine the smallest k satisfying (3) or (4) evaluating the respective CDF at each step would be a better general solution, given some initial upper bound for ISRN Probability and Statistics k. Such methods (and related variants) are employed very effectively in modern commercial and research-based statistical packages. Methods to obtain provable bounds with known error signs (typically one would require to underestimate (1) and (2), whilst overestimating (3) and (4) in most engineering and computer science applications) principally include the Bernstein/Chernoff/Hoeffding-type exponential probability inequalities and their close variants [1, 2, 6, 7].
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