On computing the distribution function for the Poisson binomial distribution
The Poisson binomial distribution is the distribution of the sum of independent and non-identically distributed random indicators. Each indicator follows a Bernoulli distribution and the individual probabilities of success vary. When all success probabilities are equal, the Poisson binomial distribution is a binomial distribution. The Poisson binomial distribution has many applications in different areas such as reliability, actuarial science, survey sampling, econometrics, etc. The computing of the cumulative distribution function (cdf) of the Poisson binomial distribution, however, is not straightforward. Approximation methods such as the Poisson approximation and normal approximations have been used in literature. Recursive formulae also have been used to compute the cdf in some areas. In this paper, we present a simple derivation for an exact formula with a closed-form expression for the cdf of the Poisson binomial distribution. The derivation uses the discrete Fourier transform of the characteristic function of the distribution. We develop an algorithm that efficiently implements the exact formula. Numerical studies were conducted to study the accuracy of the developed algorithm and approximation methods. We also studied the computational efficiency of different methods. The paper is concluded with a discussion on the use of different methods in practice and some suggestions for practitioners.
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