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On the nature of the binomial distribution

Part of: Distribution theory - Probability Distribution theory

Published online by Cambridge University Press:  14 July 2016

P. Vellaisamy  and
Abraham P. Punnen
P. Vellaisamy*
Affiliation:
Indian Institute of Technology, Bombay
Abraham P. Punnen
Affiliation:
University of New Brunswick
*
∗ Postal address: Department of Mathematics, Indian Institute of Technology, Mumbai-400 076, India. Email address: pv@math.iitb.ernet.in
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Abstract

We examine how the binomial distribution B(n,p) arises as the distribution Sn = ∑i=1nXi of an arbitrary sequence of Bernoulli variables. It is shown that B(n,p) arises in infinitely many ways as the distribution of dependent and non-identical Bernoulli variables, and arises uniquely as that of independent Bernoulli variables. A number of illustrative examples are given. The cases B(2,p) and B(3,p) are completely analyzed to bring out some of the intrinsic properties of the binomial distribution. The conditions under which Sn follows B(n,p), given that Sn-1 is not necessarily a binomial variable, are investigated. Several natural characterizations of B(n,p), including one which relates the binomial distributions and the Poisson process, are also given. These results and characterizations lead to a better understanding of the nature of the binomial distribution and enhance the utility.

Keywords

sums of Bernoulli variablesbinomial distributionstatistical independencecharacterizationsrandom sumPoisson distributionPoisson process

MSC classification

Primary: 60E05: Distributions
Secondary: 62E10: Characterization and structure theory 62E15: Exact distribution theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 1 , March 2001 , pp. 36 - 44
Copyright
Copyright © by the Applied Probability Trust 2001 

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Footnotes

∗∗

Current address: Department of Mathematics, Indian Institute of Technology, Kanpur-208016, India.

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