## Tight Probability Bounds with Pairwise Independence

 While useful probability bounds for $$n$$ pairwise independent Bernoulli random variables adding up to at least an integer $$k$$ have been proposed in the literature, none of these bounds are tight in general. In this paper, we provide several results in this direction. Firstly, when $$k = 1$$, the tightest upper bound on the … Read more

## Extremal Probability Bounds in Combinatorial Optimization

In this paper, we compute the tightest possible bounds on the probability that the optimal value of a combinatorial optimization problem in maximization form with a random objective exceeds a given number, assuming only knowledge of the marginal distributions of the objective coefficient vector. The bounds are “extremal” since they are valid across all joint … Read more

## Tree Bounds for Sums of Bernoulli Random Variables: A Linear Optimization Approach

We study the problem of computing the tightest upper and lower bounds on the probability that the sum of n dependent Bernoulli random variables exceeds an integer k. Under knowledge of all pairs of bivariate distributions denoted by a complete graph, the bounds are NP-hard to compute. When the bivariate distributions are specified on a … Read more

## Tight tail probability bounds for distribution-free decision making

Chebyshev’s inequality provides an upper bound on the tail probability of a random variable based on its mean and variance. While tight, the inequality has been criticized for only being attained by pathological distributions that abuse the unboundedness of the underlying support and are not considered realistic in many applications. We provide alternative tight lower … Read more

## Chebyshev Inequalities for Products of Random Variables

We derive sharp probability bounds on the tails of a product of symmetric non-negative random variables using only information about their first two moments. If the covariance matrix of the random variables is known exactly, these bounds can be computed numerically using semidefinite programming. If only an upper bound on the covariance matrix is available, … Read more

## Optimal Response to Epidemics and Cyber Attacks in Networks

This paper introduces novel formulations for optimally responding to epidemics and cyber attacks in networks. In our models, at a given time period, network nodes (e.g., users or computing resources) are associated with probabilities of being infected, and each network edge is associated with some probability of propagating the infection. A decision maker would like … Read more