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