Lower bounds on the probability of a finite union of events

Department of Mathematical Sciences

Research output: Contribution to journal › Journal article › Research › peer-review

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Lower bounds on the probability of a finite union of events. / Yang, Jun; Alajaji, Fady; Takahara, Glen.

In: SIAM Journal on Discrete Mathematics, Vol. 30, No. 3, 2016, p. 1437-1452.

Research output: Contribution to journal › Journal article › Research › peer-review

Harvard

Yang, J, Alajaji, F & Takahara, G 2016, 'Lower bounds on the probability of a finite union of events', SIAM Journal on Discrete Mathematics, vol. 30, no. 3, pp. 1437-1452. https://doi.org/10.1137/15M100866X

APA

Yang, J., Alajaji, F., & Takahara, G. (2016). Lower bounds on the probability of a finite union of events. SIAM Journal on Discrete Mathematics, 30(3), 1437-1452. https://doi.org/10.1137/15M100866X

Vancouver

Yang J, Alajaji F, Takahara G. Lower bounds on the probability of a finite union of events. SIAM Journal on Discrete Mathematics. 2016;30(3):1437-1452. https://doi.org/10.1137/15M100866X

Author

Yang, Jun ; Alajaji, Fady ; Takahara, Glen. / Lower bounds on the probability of a finite union of events. In: SIAM Journal on Discrete Mathematics. 2016 ; Vol. 30, No. 3. pp. 1437-1452.

Bibtex

@article{e2536f12933d49768d1049eef19af6f8,

title = "Lower bounds on the probability of a finite union of events",

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.",

keywords = "Linear programming, Lower and upper bounds, Optimal bounds, Probability of a finite union of events",

author = "Jun Yang and Fady Alajaji and Glen Takahara",

note = "Publisher Copyright: Copyright {\textcopyright} by SIAM.",

year = "2016",

doi = "10.1137/15M100866X",

language = "English",

volume = "30",

pages = "1437--1452",

journal = "SIAM Journal on Discrete Mathematics",

issn = "0895-4801",

publisher = "Society for Industrial and Applied Mathematics",

number = "3",

}

RIS

TY - JOUR

T1 - Lower bounds on the probability of a finite union of events

AU - Yang, Jun

AU - Alajaji, Fady

AU - Takahara, Glen

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