Closed-Form Expression for the Poisson-Binomial Probability Density Function

The Poisson-binomial probability density function (pdf) describes the numbers of successes in N independent trials, when the individual probabilities of success vary across trials. Its use is pervasive in applications, such as fault tolerance, signal detection, target tracking, object classification/identification, multi-sensor data fusion, system management, and performance characterization, among others. We present a closed-form expression for this pdf, and we discuss several of its advantages regarding computing speed and implementation and in simplifying analysis, with examples of the latter including the computation of moments and the development of new trigonometric identities for the binomial coefficient and the binomial cumulative distribution function (cdf). Finally we also pose and address the inverse Poisson-binomial problem; that is, given such pdf, how to find (within a permutation) the probabilities of success of the individual trials.