A sharp upper probability bound for the occurrence of at least m out of n events

Abstract

Consider a fixed set of n events. Let S 1 denote the sum of the probabilities of occurrence of each event and S 2 the sum of the probabilities of occurrence of each of the pairs of events. Using a dual linear programming method a sharp upper bound is derived for the probability of occurrence of at least m out of the n events when S 1 and S 2 are given. The bound improves one of the inequalities obtained by Sathe et al. (1980).

Bonferroni-type inequalities; Chebyshev-type inequalities for the distributions on [0, n]

“…Theorems 3.2 and 3.3 in the case r = 2 leads to the results by Kwerel (1975a), Sathe et al (1980) and Platz (1985). The derivation is simple and is omitted.…”

A nested hierarchy of second order upper bounds on system failure probability

“…Galambos [30] also found the same upper bound based on first two binomial moments using a different technique. A few years later, Galambos and Mucci [31] and Platz [32] developed bounds using LP that use higher binomial moments. Prekopa in his series of papers [33][34][35][36] formulated the Bonferri Inequalities of Dawson and Sankoff [24] as a linear programming problem, replaced the first and second order probabilities with the first 𝑚 binomial moments of the random variable and obtained sharper bounds.…”

Deterministic approximations of probability inequalities

“…'s -yield sharp bounds, see for example Bahadur and Rao (1960), Bennett (1962), Chernoff (1952), Dawson and Sankoff (1967), Dupacova (1980Dupacova ( , 1987, Galambos (1977), Godwin (1955), Hoeffding (1963), Huang, Ziemba and Ben-Tal (1977), Kall and Stoyan (1982), Kankova (1977), Klein Hanev.eld (1985, Kwerel (1975), Madansky (1960), M6ri and Sz6kely (1985), Okamoto (1958), Percus and Percus (1985), Platz (1985), Pint6r (1985), Prohorov (1959), Sathe, Pradhan and Shah (1980), Seppala (1975), Sinha (1963), Sz~ntai (1985) and Yudin (1980). In this paper -based on several cited references -mean/ range, mean/variance and mean/variance/range approximations will be given.…”