Lower bounds on the probability of a finite union of events
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Lower bounds on the probability of a finite union of events. / Yang, Jun; Alajaji, Fady; Takahara, Glen.
I: SIAM Journal on Discrete Mathematics, Bind 30, Nr. 3, 2016, s. 1437-1452.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - Lower bounds on the probability of a finite union of events
AU - Yang, Jun
AU - Alajaji, Fady
AU - Takahara, Glen
N1 - Publisher Copyright: Copyright © by SIAM.
PY - 2016
Y1 - 2016
N2 - 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.
AB - 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.
KW - Linear programming
KW - Lower and upper bounds
KW - Optimal bounds
KW - Probability of a finite union of events
UR - http://www.scopus.com/inward/record.url?scp=84990944628&partnerID=8YFLogxK
U2 - 10.1137/15M100866X
DO - 10.1137/15M100866X
M3 - Journal article
AN - SCOPUS:84990944628
VL - 30
SP - 1437
EP - 1452
JO - SIAM Journal on Discrete Mathematics
JF - SIAM Journal on Discrete Mathematics
SN - 0895-4801
IS - 3
ER -
ID: 362747442