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On the nature of the binomial distribution

Part of: Distribution theory Distribution theory - Probability

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 S n = ∑i=1 n X i 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 S n follows B(n,p), given that S n-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

Information

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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