Computing best-possible bounds for the distribution of a sum of several variables is NP-hard

In many real-life situations, we know the probability distribution of two random variables x1 and x2, but we have no information about the correlation between x1 and x2; what are the possible probability distributions for the sum x1+x2? This question was originally raised by A. N. Kolmogorov. Algorithms exist that provide best-possible bounds for the distribution of x1 + x2; these algorithms have been implemented as a part of the efficient software for handling probabilistic uncertainty. A natural question is: what if we have several (n > 2) variables with known distribution, we have no information about their correlation, and we are interested in possible probability distribution for the sum y = x1 + . . . + xn? Known formulas for the case n = 2 can be (and have been) extended to this case. However, as we prove in this paper, not only are these formulas not bestpossible anymore, but in general, computing the best-possible bounds for arbitrary n is an NP-hard (computationally intra...

• Type: Journal
• Year: 2006
• Where: IJAR
• Authors: Vladik Kreinovich, Scott Ferson
• Keywords: IJAR 2006, Possible Probability Distribution, Probability Distribution, Real-life Situations