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Improved bivariate Bonferroni-type inequalities

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Abstract

Let A1, A2, ..., Am and B1, B2, ..., Bn be two sequences of events on the same probability space. Let Xm and Yn, respectively, be the numbers of those Ai and Bj which occur. Improved upper bounds and lower bounds of yl,l = P(Xm(A)[greater-or-equal, slanted] 1, Yn(B) [greater-or-equal, slanted] 1) in terms of the bivariate binomial moments Si,j are obtained, using the method of indicators. We also obtained lower bounds on yi,j = P(Xm(A)[greater-or-equal, slanted]i, Yn(B)[greater-or-equal, slanted]j).

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