Strengthened bounds for the probability of \(k\)-out-of-\(n\) events
type: Item
instance of: scholarly article
title: Strengthened bounds for the probability of \(k\)-out-of-\(n\) events
zbMATH Open document ID: 1397.60022
author: Feng Qiu
author: S. Ahmed
author: Santanu S. Dey
published in: Discrete Applied Mathematics
publication date: 2015-12-07T00:00:00Z
review text: Let \(\{A_j\}_{1\leq j\leq n}\), \(n\geq 1,\) be a set of events defined on a probability space. Define the random variables \(X_j:A_j\to\{0,1\}\) as \(X_j=1\) if \(A_j\) occurs, and \(X_j=0\) otherwise. Let \(\mu=\sum_j X_j\). Clearly, \(\mathbb{P}(\mu\geq k)\) is the probability that at least \(k\) events \(A_j\) occur. It is known that if marginal probabilities and all joint probabilities up to \(m\) (\(m\leq n-1\)) are provided, then only lower and upper bounds for \(\mathbb{P}(\mu\geq k)\) are available. A strong bound is also known only for \(\mathbb{P}(\mu\geq 1)\). Its computation is based on a polynomial size linear program. From the authors' abstract: ``In this work, we propose inequalities that can be added to this linear program to strengthen the bounds.We also show that with a slight modification of the objective function this linear program and the inequalities can be used for the more general case where \(k\) is any positive integer less than or equal to \(n\).'' Various examples are also discussed.
Mathematics Subject Classification ID: 60C05
Mathematics Subject Classification ID: 90C05
zbMATH DE Number: 6516936
zbMATH Keywords: \(k\)-of-\(n\) event
zbMATH Keywords: linear programming
zbMATH Keywords: probability bound
reviewed by: Ljuben Mutafchiev
MaRDI profile type: Publication
full work available at URL: j.dam.2015.05.023
OpenAlex ID: W2212305991
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DOI: 10.1016/J.DAM.2015.05.023
description: scientific article; zbMATH DE number 6516936