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Deterministic approximations of probability inequalities

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Abstract

A simple general framework for derivingexplicit deterministic approximations of probability inequalities of the formP(ξ⩾a) ⩽ α is presented. These approximations are based on limited parametric information about the involved random variables (such as their mean, variance, range or upper bound values). First the case of a single random variableξ is analysed, followed by the cases of independent and dependent summands\(\xi = \mathop \sum \limits_1^n \xi _i \). As examples of possible applications, a stochastic extension of the “knapsack problem” and the stochastic linear programming problem with separate chance-constraints are investigated: we provide approximate deterministic surrogates for these problems.

Zusammenfassung

Es wird ein Rahmen zur Ableitung expliziter deterministischer Approximation für Wahrscheinlichkeitsungleichungen der FormP(ξ⩾a)⩽ α angegeben. Diese Approximationen basieren auf begrenzter parametrischer Information über die beteiligten Zufallsvariablen (wie ihr Erwartungswert, Varianz, Wertebereich oder obere Schranken). Zuerst wird der Fail einer Zufallsvariablenξ analysiert, sodann werden Summen von unabhängigen Summanden\(\xi = \mathop \sum \limits_{i = 1}^n \xi _i \) betrachtet. Als Beispiele für mögliche Anwendungen wird eine stochastische Erweiterung des Rucksack-problems untersucht sowie stochastische lineare Programme mit separablen Wahrscheinlichkeitsrestriktionen. Für diese Probleme werden näherungsweise deterministische Ersatzprobleme angegeben.

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Authors and Affiliations

  1. Research Center for Water Resources Development (VITUKI), P.O. Box 27, 1453, Budapest, Hungary

    J. Pintér

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  1. J. Pintér
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Pintér, J. Deterministic approximations of probability inequalities. ZOR - Methods and Models of Operations Research 33, 219–239 (1989). https://doi.org/10.1007/BF01423332

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  • DOI: https://doi.org/10.1007/BF01423332

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