Strengthened bounds for the probability of \(k\)-out-of-\(n\) events
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.
scientific article; zbMATH DE number 6516936