# Unimodality, Independence Lead to NP-Hardness of Interval Probability Problems

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## Abstract

In many real-life situations, we only have partial information about probabilities. This information is usually described by bounds on moments, on probabilities of certain events, etc. –i.e., by characteristics *c(p)* which are linear in terms of the unknown probabilities *p* _{j}. If we know interval bounds on some such characteristics \( \underline{a}_i\leq c_i(p)\leq \bar{a}_i \), and we are interested in a characteristic *c(p)*, then we can find the bounds on *c(p)* by solving a linear programming problem.

In some situations, we also have additional conditions on the probability distribution –e.g., we may know that the two variables *x* _{1} and *x* _{2} are independent, or that the joint distribution of *x* _{1} and *x* _{2} is unimodal. We show that adding each of these conditions makes the corresponding interval probability problem NP-hard.

## Keywords

Linear Programming Problem Linear Constraint Stochastic Dominance Unimodal Distribution Integer Point## Preview

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