Skip to main content
SpringerLink
Log in

Renewed Looks at the Distribution of a Sum of Independent or Dependent Discrete Random Variables and Related Problems

  • Published:
Methodology and Computing in Applied Probability Aims and scope Submit manuscript

Abstract

Butler and Stephens (2017) have investigated the exact and approximate distributions of a sum S of independent binomial random variables with different probabilities. They used a convolution approach to find the exact distribution, whereas they heavily used the moments and cumulants to find approximations. We propose two different approaches. First, we show that the moment generating function (MGF) approach is easier and simpler to implement for finding the exact distribution of S. We also provide approximations for the distribution of S using large-scale computer simulations based on one million independent replications each. Such exact and approximate distributions are very close to analogous values reported in Butler and Stephens (2017). We show versatility of our approaches by including both exact and approximate distributions for the sum S of independent multinomial, geometric and other discrete random variables.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Billingsley P (1995) Probability and measures, 3rd edn. Wiley, New York. Anniversary Edition (2012). Wiley, New Jersey

    Google Scholar 

  • Butler K, Stephens MA (2017) The distribution of a sum of independent binomial random variables. Methodol Comput Appl Probab. https://doi.org/10.1007/s11009-016-9533-4

  • Casella G, Berger RL (1990) Statistical inference. Belmont, Duxbury

    MATH  Google Scholar 

  • Chung KL (1974) A course in probability theory. Academic Press, New York

    MATH  Google Scholar 

  • Feller W (1971) An introduction to probability theory and its applications, vol 2, 2nd edn. Wiley, New York

    MATH  Google Scholar 

  • Gut A (2003) On the moment problem for random sums. J Appl Probab 40:797–802

    Article  MathSciNet  MATH  Google Scholar 

  • Heyde CC (1963) On a property of the lognormal distribution. J Roy Stat Soc Ser B 25:392–393

    MathSciNet  MATH  Google Scholar 

  • Mukhopadhyay N (2000) Probability and statistical inference. Dekker, New York

    MATH  Google Scholar 

  • Mukhopadhyay N (2006) Introductory statistical inference. Chapman & Hall/CRC, Boca Raton

    Book  MATH  Google Scholar 

  • Rao CR (1973) Linear statistical inference and its applications, 2nd edn. Wiley, New York

    Book  Google Scholar 

  • R Core Team (2014) R: a Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna

    Google Scholar 

  • Stoyanov JM (2012) Inference problems involving moment determinacy of distributions. Comm Stat - Theory Meth 41:2864–2878

    Article  MathSciNet  MATH  Google Scholar 

  • Wolfram Research Inc (2017) Mathematica, Version 11.1, Champaign

Download references

Acknowledgments

We received a great set of comments and critics from two anonymous reviewers and one associate editor. Those substantial reports have truly helped us in strengthening our original work. We earnestly thank them as well as the Editor-in-Chief for all the help.

Author information

Authors and Affiliations

  1. Department of Statistics, University of Connecticut, Austin Building U-4120, 215 Glenbrook Road, Storrs, CT, 06269, USA

    Nitis Mukhopadhyay

  2. Department of Statistics and Applied Probability, University of California, Santa Barbara, Santa Barbara, CA, USA

    Sudeep R. Bapat

Authors
  1. Nitis Mukhopadhyay
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sudeep R. Bapat
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nitis Mukhopadhyay.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mukhopadhyay, N., Bapat, S.R. Renewed Looks at the Distribution of a Sum of Independent or Dependent Discrete Random Variables and Related Problems. Methodol Comput Appl Probab 21, 853–873 (2019). https://doi.org/10.1007/s11009-018-9690-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11009-018-9690-8

Keywords

Mathematics Subject Classiffication (2010)

Search

Navigation