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On Mohan's Property of the Poisson Distribution
Author(s) -
Panaretos John
Publication year - 1983
Publication title -
biometrical journal
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.108
H-Index - 63
eISSN - 1521-4036
pISSN - 0323-3847
DOI - 10.1002/bimj.19830250108
Subject(s) - negative binomial distribution , mathematics , hypergeometric distribution , poisson distribution , compound poisson distribution , binomial (polynomial) , combinatorics , distribution (mathematics) , property (philosophy) , random variable , hypergeometric function , poisson binomial distribution , characterization (materials science) , conditional probability distribution , integer (computer science) , binomial distribution , poisson regression , statistics , pure mathematics , mathematical analysis , beta binomial distribution , physics , population , philosophy , demography , epistemology , sociology , computer science , optics , programming language
Two interesting results encountered in the literature concerning the Poisson and the negative binomial distributions are due to Moran (1952) and Patil & Seshadri (1964), respectively. Morans result provided a fundamental property of the Poisson distribution. Roughly speaking, he has shown that if Y, Z are independent, non‐negative, integer‐valued random variables with X = Y | Z then, under some mild restrictions, the conditional distribution of Y | X is binomial if and only if Y , Z are Poisson random variables. Motivated by Morans result Patil & Seshadri obtained a general characterization. A special case of this characterization suggests that, with conditions similar to those imposed by Moran, Y | X is negative hypergeometric if and only if Y , Z are negative binomials. In this paper we examine the results of Moran and Patil & Seshadri in the case where the conditional distribution of Y | X is truncated at an arbitrary point k – 1 ( k = 1, 2, …). In fact we attempt to answer the question as to whether Morans property of the Poisson distribution, and subsequently Patil & Seshadris property of the negative binomial distribution, can be extended, in one form or another, to the case where Y | X is binomial truncated at k – 1 and negative hypergeometric truncated at k – 1 respectively.