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The Distribution of a Sum of Independent Binomial Random Variables

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Abstract

The distribution of a sum S of independent binomial random variables, each with different success probabilities, is discussed. An efficient algorithm is given to calculate the exact distribution by convolution. Two approximations are examined, one based on a method of Kolmogorov, and another based on fitting a distribution from the Pearson family. The Kolmogorov approximation is given as an algorithm, with a worked example. The Kolmogorov and Pearson approximations are compared for several given sets of binomials with different sample sizes and probabilities. Other methods of approximation are discussed and some compared numerically. The Kolmogorov approximation is found to be extremely accurate, and the Pearson curve approximation useful if extreme accuracy is not required.

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Authors and Affiliations

  1. Department of Computer and Mathematical Sciences, University of Toronto Scarborough, 1265 Military Trail, Toronto, Ontario, M1C 1A4, Canada

    Ken Butler

  2. Department of Statistics and Actuarial Sciences, Simon Fraser University, 8888 University Drive, Burnaby, British Columbia, V5A 1S6, Canada

    Michael A. Stephens

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  1. Ken Butler
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  2. Michael A. Stephens
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Correspondence to Michael A. Stephens.

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Butler, K., Stephens, M.A. The Distribution of a Sum of Independent Binomial Random Variables. Methodol Comput Appl Probab 19, 557–571 (2017). https://doi.org/10.1007/s11009-016-9533-4

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  • DOI: https://doi.org/10.1007/s11009-016-9533-4

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