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A characterization of the geometric distribution

Published online by Cambridge University Press:  14 July 2016

M. Sreehari
M. Sreehari*
Affiliation:
M. S. University, Baroda
*
∗ Postal address: Department of Statistics, Faculty of Science, M. S. University, Baroda, 390 002, India.
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Abstract

Let X1, X2, …, Xn be independent identically distributed positive integer-valued random variables with order statistics X1:n, X2:n, …, Xn:n. We prove that if the random variable X2:n – X1:n is independent of the events [X1:n = m] and [X1:n = k], for fixed k > m > 1, then the Xi's are geometric. This is related to a characterization problem raised by Arnold (1980).

Keywords

ORDER STATISTICS
Type
Short Communications
Information
Journal of Applied Probability , Volume 20 , Issue 1 , March 1983 , pp. 209 - 212
Copyright
Copyright © Applied Probability Trust 1983 

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Footnotes

Research carried out while the author was at Southern Illinois University at Carbondale.

References

Arnold, B. C. (1980) Two characterizations of the geometric distribution. J. Appl. Prob. 17, 570–573.CrossRefGoogle Scholar
Sreehari, M. (1981) Two more similarities between the exponential and the geometric distributions. Unpublished.Google Scholar
Wong, W. Y. (1980) On set with “lack of memory” property and some applications to the characterization of geometric distribution. Bull. Malaysian Math. Soc. (2) 3, 29–31.Google Scholar