Closed form two-sided bounds for probabilities that at least r and exactly r out of n events occur

Closed form two-sided bounds for probabilities that at least r and exactly r out of n events occur

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Article ID: iaor19881200
Country: United States
Volume: 14
Issue: 2
Start Page Number: 317
End Page Number: 342
Publication Date: May 1989
Journal: Mathematics of Operations Research
Authors: ,
Keywords: programming: linear
Abstract:

In two previous papers Prékopa gave algorithms to approximate probabilities that at least r and exactly r out of n events occurs (1•rn). Primal and dual linear programming problems were formulated and solved by dual type algorithms. The purpose of the present paper is to give closed forms for the basis inverse and the corresponding dual vector in case of an arbitrary basis, furthermore to give closed forms for the lower and upper bounds, approximating the probability in question, in case of a dual feasible basis. In the case when the probability that at least one out of n events occurs is approximated, it is shown that the absolute values of the components of any dual vector form a monotonically decreasing sequence. The paper improves the method of inclusion-exclusion, proves new probability inequalities and proves the sharpness of some known inequalities.

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