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A CHARACTERIZATION OF GEOMETRIC DISTRIBUTIONS THROUGH CONDITIONAL INDEPENDENCE
Author(s) -
Liang TaChen,
Balakrishnan N.
Publication year - 1993
Publication title -
australian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 0004-9581
DOI - 10.1111/j.1467-842x.1993.tb01328.x
Subject(s) - event (particle physics) , random variable , conditional independence , characterization (materials science) , mathematics , geometric distribution , independence (probability theory) , combinatorics , conditional probability distribution , integer (computer science) , conditional probability , discrete mathematics , probability distribution , statistics , physics , computer science , quantum mechanics , optics , programming language
Summary Let X 1 ,…, X n be mutually independent non‐negative integer‐valued random variables with probability mass functions f i ( x ) > 0 for z = 0,1,…. Let E denote the event that { X 1 ≥ X 2 ≥…≥ X n }. This note shows that, conditional on the event E, X i ‐ X i + 1 and X i + 1 are independent for all t = 1,…, k if and only if X i ( i = 1,…, k ) are geometric random variables, where 1 ≤ k ≤ n ‐1. The k geometric distributions can have different parameters θ i , i = 1,…, k.