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PROBABILITY INEQUALITIES FOR SUMS OF BOUNDED RANDOM VARIABLES
, 1962
"... Upper bounds are derived for the probability that the sum S of n independent random variables exceeds its mean ES by a positive number nt. It is assumed that the range of each summand of S is bounded or bounded above. The bounds for Pr(SES> nt) depend only on the endpoints of the ranges of the smum ..."
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Cited by 1498 (2 self)
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Upper bounds are derived for the probability that the sum S of n independent random variables exceeds its mean ES by a positive number nt. It is assumed that the range of each summand of S is bounded or bounded above. The bounds for Pr(SES> nt) depend only on the endpoints of the ranges of the smumands and the mean, or the mean and the variance of S. These results are then used to obtain analogous inequalities for certain sums of dependent random variables such as U statistics and the sum of a random sample without replacement from a finite population.
Sand Report
, 2001
"... The DAKOTA (Design Analysis Kit for Optimization and Terascale Applications) toolkit provides a flexible and extensible interface between simulation codes and iterative analysis methods. DAKOTA contains algorithms for optimization with gradient and nongradientbased methods; uncertainty quantificati ..."
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The DAKOTA (Design Analysis Kit for Optimization and Terascale Applications) toolkit provides a flexible and extensible interface between simulation codes and iterative analysis methods. DAKOTA contains algorithms for optimization with gradient and nongradientbased methods; uncertainty quantification with sampling, analytic reliability, and stochastic finite element methods; parameter estimation with nonlinear least squares methods; and sensitivity analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogatebased optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing objectoriented design to implement abstractions of the key components required for iterative systems analyses, the DAKOTA toolkit provides a flexible and extensible problemsolving environment for design and performance analysis of computational models on high performance computers.
Constructing Probability Boxes and . . .
, 2003
"... This report summarizes a variety of the most useful and commonly applied methods for obtaining DempsterShafer structures, and their mathematical kin probability boxes, from empirical information or theoretical knowledge. The report includes a review of the aggregation methods for handling agreement ..."
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This report summarizes a variety of the most useful and commonly applied methods for obtaining DempsterShafer structures, and their mathematical kin probability boxes, from empirical information or theoretical knowledge. The report includes a review of the aggregation methods for handling agreement and conflict when multiple such objects are obtained from different sources.
Full length article On Chebyshev–Markov–Krein inequalities
, 2012
"... We review the topic of Chebyshev–Markov–Krein inequalities, i.e. estimates for inf f dν and sup f dν ν∈V (µ) ν∈V (µ) where µ is a nonnegative finite measure, and V (µ) is the set of all nonnegative finite measures ν satisfying u dν = u dµ for all u ∈ U, where U is a finitedimensional subspace. Fo ..."
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We review the topic of Chebyshev–Markov–Krein inequalities, i.e. estimates for inf f dν and sup f dν ν∈V (µ) ν∈V (µ) where µ is a nonnegative finite measure, and V (µ) is the set of all nonnegative finite measures ν satisfying u dν = u dµ for all u ∈ U, where U is a finitedimensional subspace. For U a finitedimensional Tspace on [a, b], we prove correct necessary and sufficient conditions for when a given nonnegative function f ∈ C[a, b] satisfies � ξ − � ξ − � ξ+ � ξ+ f dµξ ≤ f dν ≤ f dν ≤ f dµξ a a