Abstract
In this paper, lower bounds on the probability of a finite union of events are considered, i.e., P(UNi=1Ai, in terms of the individual event probabilities {P(Ai), i = 1, ... ,N} and the sums of the pairwise event probabilities, i.e., {Σj:j≠i P(Ai ∩ Aj), i = 1, ... ,N}. The contribution of this paper includes the following: (i) in the class of all lower bounds that are established in terms of only the P(Ai)'s and σj:j≠ij P(Ai ≠ Aj)'s, the optimal lower bound is given numerically by solving a linear programming (LP) problem with N2 - N + 1 variables; (ii) a new analytical lower bound is proposed based on a relaxed LP problem, which is at least as good as the bound due to Kuai, Alajaji, and Takahara [Discrete Math., 215 (2000), pp. 147-158]; (iii) numerical examples are provided to illustrate the performance of the bounds.
Original language | English |
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Journal | SIAM Journal on Discrete Mathematics |
Volume | 30 |
Issue number | 3 |
Pages (from-to) | 1437-1452 |
Number of pages | 16 |
ISSN | 0895-4801 |
DOIs | |
Publication status | Published - 2016 |
Externally published | Yes |
Bibliographical note
Publisher Copyright:Copyright © by SIAM.
Keywords
- Linear programming
- Lower and upper bounds
- Optimal bounds
- Probability of a finite union of events