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Further studies of bivariate Bonferroni-type inequalities

Part of: Distribution theory - Probability Combinatorial probability

Published online by Cambridge University Press:  14 July 2016

Janos Galambos  and
Min-Young Lee
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Abstract

Let A1, A2, …, An and B1, B2,. ., BN be two sequences of events on the same probability space. Let mn(A) and mN(B), respectively, be the number of those Aj and Bj which occur. Let Si,j denote the joint ith binomial moment of mn(A) and jth binomial moment of mN(B), 0 ≤ i ≤ n, 0 ≤ j ≤ N. For fixed non-negative integers a and b, we establish both lower and upper bounds on the distribution P(mn(A) = r, mN(B) = u) by linear combinations of Si,j, 0 ≤ i ≤ a, 0 ≤ j ≤ b. When both a and b are even, all mentioned S¡,j are utilized in both the upper and the lower bound. In a set of remarks the results are analyzed and their relation to the existing literature, including the univariate case, is discussed.

Keywords

BIVARIATE DISTRIBUTIONNUMBER OF OCCURRENCESBIVARIATE BINOMIAL MOMENTS

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60C05: Combinatorial probability
Type
Part 2 Probabilistic Methods
Information
Journal of Applied Probability , Volume 31 , Issue A , 1994 , pp. 63 - 69
Copyright
Copyright © Applied Probability Trust 1994 

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References

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