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On a characterization of Poisson distributions

Published online by Cambridge University Press:  14 July 2016

J. Aczél
J. Aczél*
Affiliation:
University of Waterloo, Ontario
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Abstract

The conjecture pronounced at the end of the paper of Srivastava and Srivastava (1970) is proved in this paper. It gives the following characterization of (bivariate) Poisson distributions. Suppose that items of two types have been observed certain numbers of times, but these original observations have been reduced due to a destructive process which is the product of two binomial distributions and that the probabilities of these reduced numbers are the same whether damaged or undamaged. Then the original random variables had a bivariate Poisson distribution with zero mutual dependence coefficient.

Keywords

BIVARIATE POISSON DISTRIBUTIONSBINOMIAL DISTRIBUTIONSCAUCHY'S BIVARIATE FUNCTIONAL EQUATIONEXTENSION OF SOLUTIONSCONTINUITYMEASURABILITYBOUNDEDNESS
Type
Short Communications
Information
Journal of Applied Probability , Volume 9 , Issue 4 , December 1972 , pp. 852 - 856
Copyright
Copyright © Applied Probability Trust 1972 

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References

Aczél, J. (1966) Lectures on Functional Equations and their Applications. Academic Press, New York and London.Google Scholar
Daróczy, Z. and Losonczi, L. (1967) Über die Erweiterung der auf einer Punktmenge additiven Funktionen. Publ. Math. Debrecen 14, 239–245.Google Scholar
Srivastava, R. C. and Srivastava, A. B. L. (1970) On a characterization of Poisson distribution. J. Appl. Prob. 7, 497–501.Google Scholar
Talwalker, Sh. (1970) A characterization of the double Poisson distribution. Sankhya Ser. A. 32, 265–270.Google Scholar