Probabilistic Inequalities

  • Michael Zabarankin
  • Stan Uryasev
Part of the Springer Optimization and Its Applications book series (SOIA, volume 85)


In various statistical decision problems dealing with safety and reliability, risk is often interpreted as the probability of a dread event or disaster, and minimizing the probability of a highly undesirable event is known as the safety-first principle [50]. If the CDF of X is either unknown or complex, the probability in question can be estimated through more simple characteristics such as mean and standard deviation of X, for example, by Markov’s and Chebyshev’s inequalities. Also, if the probability depends on decision variables, then, in general, an optimization problem, in which it is either minimized or constrained, is nonconvex. In this case, the probability can be estimated by an appropriate probabilistic inequality, and then the optimization problem can be approximated by a convex one; see, e.g., [3, 32].


Probability Inequalities Safety-first Principle Chebyshev Dread Event Statistical Decision Problem 
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Copyright information

© Springer Science+Business Media, New York 2014

Authors and Affiliations

  • Michael Zabarankin
    • 1
  • Stan Uryasev
    • 2
  1. 1.Department of Mathematical SciencesStevens Institute of TechnologyHobokenUSA
  2. 2.Department of Industrial and Systems EngineeringUniversity of FloridaGainesvilleUSA

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