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The Distribution of a Sum of Independent Binomial Random Variables

Author

Listed:
  • Ken Butler

    (University of Toronto Scarborough)

  • Michael A. Stephens

    (Simon Fraser University)

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.

Suggested Citation

  • Ken Butler & Michael A. Stephens, 2017. "The Distribution of a Sum of Independent Binomial Random Variables," Methodology and Computing in Applied Probability, Springer, vol. 19(2), pages 557-571, June.
    • Handle: RePEc:spr:metcap:v:19:y:2017:i:2:d:10.1007_s11009-016-9533-4
    DOI: 10.1007/s11009-016-9533-4
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    References listed on IDEAS

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    1. Sah, Raaj Kumar, 1989. "Comparative properties of sums of independent binomials with different parameters," Economics Letters, Elsevier, vol. 31(1), pages 27-30.
    2. Rob Eisinga & Manfred Te Grotenhuis & Ben Pelzer, 2013. "Saddlepoint approximations for the sum of independent non-identically distributed binomial random variables," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 67(2), pages 190-201, May.
    3. James Benneyan & Aysun Taşeli, 2010. "Exact and approximate probability distributions of evidence-based bundle composite compliance measures," Health Care Management Science, Springer, vol. 13(3), pages 193-209, September.
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    Cited by:

    1. Nitis Mukhopadhyay & Sudeep R. Bapat, 2019. "Renewed Looks at the Distribution of a Sum of Independent or Dependent Discrete Random Variables and Related Problems," Methodology and Computing in Applied Probability, Springer, vol. 21(3), pages 853-873, September.
    2. Vivek Verma & Dilip C. Nath, 2019. "Characterization Of The Sum Of Binomial Random Variables Under Ranked Set Sampling," Statistics in Transition New Series, Polish Statistical Association, vol. 20(3), pages 1-29, September.
    3. repec:exl:29stat:v:20:y:2019:i:3:p:1-30 is not listed on IDEAS
    4. Baena-Mirabete, S. & Puig, P., 2020. "Computing probabilities of integer-valued random variables by recurrence relations," Statistics & Probability Letters, Elsevier, vol. 161(C).
    5. Verma Vivek & Nath Dilip C., 2019. "Characterization Of The Sum Of Binomial Random Variables Under Ranked Set Sampling," Statistics in Transition New Series, Polish Statistical Association, vol. 20(3), pages 1-29, September.

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