Tight Probability Bounds with Pairwise Independence

Abstract

While some useful probability bounds for the sum of n pairwise independent Bernoulli random variables exceeding an integer k have been proposed in the literature, none of these bounds are tight in general. In this paper, we provide three results towards finding tight probability bounds for this class of problems. Firstly, when k = 1, the tightest upper bound on the probability of the union of n pairwise independent events is provided in closed form for any input marginal probability vector in [0,1]^n. Building on this, we show that the ratio of Boole's union bound and the tight pairwise independent bound is upper bounded by 4/3 and this bound is attained. Secondly, for n pairwise independent events with identical probabilities p in [0,1], the tightest upper bound on the probability that at least k occur is provided in closed form. Lastly, for non-identical pairwise independent events, new upper bounds are derived for any k ≥ 2. While these bounds are not always tight, they improve on existing bounds and in special instances are shown to be tight. The proofs of tightness are developed using techniques of linear optimization and should be of independent interest. Numerical examples are provided to illustrate when the bounds provide significant improvement over existing bounds.