Tight Probability Bounds with Pairwise Independence

Probability bounds on the sum of $n$ pairwise independent Bernoulli random variables exceeding an integer $k$ have been proposed in the literature. However, these bounds are not tight in general. In this paper, we provide three results towards finding tight probability bounds on the sum of pairwise independent Bernoulli random variables. Firstly, for $k = 1$, the tightest upper bound on the probability of the union of $n$ pairwise independent events is provided. Secondly, for $k \geq 2$, the tightest upper bound with identical marginals is provided. Lastly, for general pairwise independent Bernoulli random variables, new upper bounds are derived for $k \geq 2$, by ordering the probabilities. These bounds improve on existing bounds and are tight under certain conditions. The proofs of tightness are developed using techniques of linear optimization. Numerical examples are provided to quantify the improvement of the bounds over existing bounds.