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A characterization of the gamma distribution by the negative binomial distribution

Published online by Cambridge University Press:  14 July 2016

Jan Engel  and
Mynt Zijlstra
Jan Engel*
Affiliation:
N. V. Philips, Eindhoven
Mynt Zijlstra*
Affiliation:
N. V. Philips, Eindhoven
*
∗N. V. Philips, ISA-Quantitative Methods, Building HSM–301, 5600 MD Eindhoven, The Netherlands.
∗N. V. Philips, ISA-Quantitative Methods, Building HSM–301, 5600 MD Eindhoven, The Netherlands.
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Abstract

It is proved that for a Poisson process there exists a one-to-one relation between the distribution of the random variable N(Y) and the distribution of the non-negative random variable Y. This relation is used to characterize the gamma distribution by the negative binomial distribution. Furthermore it is applied to obtain some characterizations of the exponential distribution.

Keywords

GAMMA DISTRIBUTIONNEGATIVE BINOMIAL DISTRIBUTIONCHARACTERIZATIONPOISSON PROCESSGEOMETRIC DISTRIBUTIONEXPONENTIAL DISTRIBUTIONMOMENT PROBLEM
Type
Short Communications
Information
Journal of Applied Probability , Volume 17 , Issue 4 , December 1980 , pp. 1138 - 1144
Copyright
Copyright © Applied Probability Trust 

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References

Bosch, K. (1977) Eine Charakterisierung der Exponentialverteilungen. Z. Angew. Math. Mech. 57, 609–610.Google Scholar
Feller, W. (1967) An Introduction to Probability Theory and its Applications, Vol. 1, 3rd edn. Wiley, New York.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Galambos, J. and Kotz, S. (1978) Characterizations of Probability Distributions. Lecture Notes in Mathematics 675, Springer-Verlag, Berlin.Google Scholar
Johnson, N. L. and Kotz, S. (1969) Distributions in Statistics, Vols, 1–4. Wiley, New York.Google Scholar
Kotz, S. (1974) Characterizations of statistical distributions: a supplement to recent surveys. Internat. Statist. Rev. 42, 39–65.CrossRefGoogle Scholar
Krishnaji, N. (1971) Note on a characterizing property of the exponential distribution. Ann. Math. Statist. 42, 361–362.Google Scholar
Puri, P. A. (1973) On a property of exponential and geometric distributions and its relevance to multivariate failure rate. Sankhya 35, 61–68.Google Scholar