An Algorithm for Computing the Distribution Function of the Generalized Poisson-Binomial Distribution

The Poisson-binomial distribution is useful in many applied problems in engineering, actuarial science, and data mining. The Poisson-binomial distribution models the distribution of the sum of independent but not identically distributed Bernoulli random variables whose success probabilities vary. In this paper, we extend the Poisson-binomial distribution to the generalized Poisson-binomial (GPB) distribution. The GPB distribution is defined in cases where the Bernoulli variables can take any two arbitrary values instead of 0 and~1. The GPB distribution is useful in many areas such as voting theory, actuarial science, warranty prediction, and probability theory. With few previous works studying the GPB distribution, we derive the probability distribution via the discrete Fourier transform of the characteristic function of the distribution. We develop an efficient algorithm for computing the distribution function, which uses the fast Fourier transform. We test the accuracy of the developed algorithm upon comparing it with enumeration-based exact method and the results from the binomial distribution. We also study the computational time of the algorithm in various parameter settings. Finally, we discus the factors affecting the computational efficiency of this algorithm, and illustrate the use of the software package.