## New Bounds for the k-out-of-n Type Probabilities and Their Applications

##### Abstract

The contribution of the shape information of the underlying distribution in probability
bounding problem is investigated and an efficient linear programming based
bounding methodology, which takes advantage of the advanced optimization techniques,
probability theory, and the state-of-the-art tools, to obtain robust and
efficiently computable bounds for the probabilities that at least k and exactly
k-out-of-n events occur is developed. The k-out-of-n type probability bounding
problem is formulated as linear programs under the assumption that the probability
distribution is unimodal. The dual feasible bases structures of the relaxed
versions of linear programs involved are fully described. The bounds for the probability
that at least k and exactly k-out-of-n events occur are obtained in the form
of formulas. A dual based linear programming algorithm is proposed to obtain
bounds as the customized algorithmic solutions of the LP’s formulated. Numerical
examples are presented to show that the use of shape constraint significantly
improves on the bounds for the probabilities that at least k and exactly k-out-of-n
events occur when only first a few binomial moments are known. An application
in PERT, where the shape of the underlying probability distribution can be used
to obtain bounds for the distribution of the critical path length, is presented.