Reliability of Elastic‐Plastic Frame Structures

An effective algorithm for estimating the reliability of frame structures made of an elastic perfectly plastic material is presented. It is assumed that the structure is working in uniaxial state of stress, the material parameters and the limit values of the loads are described by random variables, all loads are acting in a static manner, the probability density functions of all random variables, which describe the structure and loads are known. As the structure reliability measure the probability of failure and corresponding reliability index are regarded. The dead (constant) load and the climatic loads are taken into account. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A comprehensive probabilistic analysis of approximate SIR-type epidemiological models via full randomized discrete-time Markov chain formulation with applications

This paper provides a comprehensive probabilistic analysis of a full randomization of approximate SIR-type epidemiological models based on discrete-time Markov chain formulation. The randomization is performed by assuming that all input data (initial conditions, the contagion, and recovering rates involved in the transition matrix) are random variables instead of deterministic constants. In the first part of the paper, we determine explicit expressions for the so called first probability density function of each subpopulation identified as the corresponding states of the Markov chain (susceptible, infected, and recovered) in terms of the probability density function of each input random variable. Afterwards, we obtain the probability density functions of the times until a given proportion of the population remains susceptible, infected, and recovered, respectively. The theoretical analysis is completed by computing explicit expressions of important randomized epidemiological quantities, namely, the basic reproduction number, the effective reproduction number, and the herd immunity threshold. The study is conducted under very general assumptions and taking extensive advantage of the random variable transformation technique. The second part of the paper is devoted to apply our theoretical findings to describe the dynamics of the pandemic influenza in Egypt using simulated data excerpted from the literature. The simulations are complemented with valuable information, which is seldom displayed in epidemiological models. In spite of the nonlinear mathematical nature of SIR epidemiological model, our results show a strong agreement with the approximation via an appropriate randomized Markov chain. A justification in this regard is discussed.     

A method for stochastic multiple criteria decision making based on pairwise comparisons of alternatives with random evaluations

This paper proposes a method for solving stochastic multiple criteria decision making (MCDM) problems, where evaluations of alternatives on considered criteria are random variables with known probability density functions or probability mass functions. Probabilities on all possible results of pairwise comparisons of alternatives are first calculated using Probability Theory. Then, all possible results of pairwise comparisons are classified into superior, indifferent and inferior ones using a predefined identification rule. Consequently, the probabilities on all possible results of pairwise comparisons are partitioned into superior, indifferent and inferior probabilities. Furthermore, based on the derived probabilities, an algorithm is developed to rank the alternatives. Finally, a numerical example is used to illustrate the feasibility and validity of the proposed method.     

二维连续型随机变量函数分布的一个定理

给出求二维连续型随机变量函数分布的一个定理,并籍以导出二维随机变量和差积商的概率密度函数公式.     

On poset similarity

Let P be a poset, and let A be an element of its strict incidence algebra. Saks (SIAM J. Algebraic Discrete Methods 1 (1980) 211–215; Discrete Math. 59 (1986) 135–166) and Gansner (SIAM J. Algebraic Discrete Methods 2 (1981) 429–440) proved that the kth Dilworth number of P is less than or equal to the dimension of the nullspace of A^k, and that there is some member of the strict incidence algebra of P for which equality is attained (for all k simultaneously). In this paper we focus attention on the question of when equality is attained with the strict zeta matrix, and proceed under a particular random poset model. We provide an invariant depending only on two measures of nonunimodality of the level structure for the poset that, with probability tending to 1 as the smallest level tends to infinity, takes on the same value as the inequality gap between the width of P and the dimension of the nullspace of its strict zeta matrix. In particular, we characterize the level structures for which the width of P is, with probability tending to 1, equal to the dimension of the nullspace of its strict zeta matrix. As a consequence, by the Kleitman–Rothschild Theorem 5, almost all posets in the Uniform random poset model have width equal to the dimension of the nullspace of their zeta matrices. We hope this is a first step toward a complete characterization of when equality holds in Saks’ and Gansner's inequality for the strict zeta matrix and for all k. New to this paper are also the canonical representatives of the poset similarity classes (where two posets are said to be similar if their strict zeta matrices are similar in the matrix-theoretic sense), and these form the setting for our work on Saks’ and Gansner's inequalities. (Also new are two functions that measure the nonunimodality of a sequence of real numbers.)     

Probability set functions

A probability set function is interpretable as a probability distribution on binary sequences of fixed length. Cumulants of probability set functions enjoy particularly simple properties which make them more manageable than cumulants of general random variables. We derive some identities satisfied by cumulants of probability set functions which we believe to be new. Probability set functions may be expanded in terms of their cumulants. We derive an expansion which allows the construction of examples of probability set functions whose cumulants are arbitrary, restricted only by their absolute values. It is known that this phenomenon cannot occur for continuous probability distributions. Some particular examples of probability set functions are considered, and their cumulants are computed, leading to a conjecture on the upper bound of the values of cumulants. Moments of probability set functions determined by arithmetical conditions are computed in a final example.Dedicated to our friend, W.A. Beyer. Financial support for this work was derived from the U.S.D.O.E. Human Genome Project, through the Center for Human Genome Studies at Los Alamos National Laboratory, and also through the Center for Nonlinear Studies, Los Alamos National Laboratory, LANL report LAUR-97-323.     

Convergence rates for weighted sums in noncommutative probability space

We study convergence rates for weighted sums of pairwise independent random variables in a noncommutative probability space of which the weights are in a von Neumann algebra. As applications, we first study convergence rates for weighted sums of random variables in the noncommutative Lorentz space, and second we study convergence rates for weighted sums of probability measures with respect to the free additive convolution.     

On the comparison of distribution functions of random variables

We prove a rather general comparison principle for the distribution functions of random variables. As a consequence, we obtain a criterion for the equivalence in distribution in the vector sense of an arbitrary sequence of random variables to the Rademacher system; we study the applications of this principle to special cases.     

Series representations for densities functions of a family of distributions—Application to sums of independent random variables

Series representations for several density functions are obtained as mixtures of generalized gamma distributions with discrete mass probability weights, by using the exponential expansion and the binomial theorem. Based on these results, approximations based on mixtures of generalized gamma distributions are proposed to approximate the distribution of the sum of independent random variables, which may not be identically distributed. The applicability of the proposed approximations are illustrated for the sum of independent Rayleigh random variables, the sum of independent gamma random variables, and the sum of independent Weibull random variables. Numerical studies are presented to assess the precision of these approximations.     

Record values with constraints

Record values are very popular in probability and mathematical statistics. There are many books and papers concerned with classical record values and record times, i.e., records in sequences of independent equally distributed random variables. In recent times, new types of record values (records in the F ^α-scheme, record values in sequences of unequally distributed random variables, records with confirmations, exceedance record values) have been proposed and examined. The present paper proposes another record scheme (so-called “records with constraint”). Various cases are studied in which these records may be useful. For these record values, we give their joint density functions and discover some of their properties. For particular cases of utmost importance, when the initial random variables are independent and have equal exponential distribution, we obtain fairly simple representations of records with constraints as sums of independent equally distributed random terms.     

A new data envelopment analysis model with fuzzy random inputs and outputs

This paper first presents several formulas for mean chance distributions of triangular fuzzy random variables and their functions, then develops a new class of fuzzy random data envelopment analysis (FRDEA) models with mean chance constraints, in which the inputs and outputs are assumed to be characterized by fuzzy random variables with known possibility and probability distributions. According to the established formulas for the mean chance distributions, we can turn the mean chance constraints into their equivalent stochastic ones. On the other hand, since the objective in the FRDEA model is the expectation about the ratio of the weighted sum of outputs and the weighted sum of inputs for a target decision-making unite (DMU), for general fuzzy random inputs and outputs, we suggest an approximation method to evaluate the objective; and for triangular fuzzy random inputs and outputs, we propose a method to reduce the objective to its equivalent stochastic one. As a consequence, under the assumption that the inputs and the outputs are triangular fuzzy random vectors, the proposed FRDEA model can be reduced to its equivalent stochastic programming one, in which the constraints contain the standard normal distribution function, and the objective is the expectation for a function of the normal random variable. To solve the equivalent stochastic programming model, we design a hybrid algorithm by integrating stochastic simulation and genetic algorithm (GA). Finally, one numerical example is presented to demonstrate the proposed FRDEA modeling idea and the effectiveness of the designed hybrid algorithm.     

A semidefinite optimization approach to the steady-state analysis of queueing systems

Obtaining (tail) probabilities from a transform function is an important topic in queueing theory. To obtain these probabilities in discrete-time queueing systems, we have to invert probability generating functions, since most important distributions in discrete-time queueing systems can be determined in the form of probability generating functions. In this paper, we calculate the tail probabilities of two particular random variables in discrete-time priority queueing systems, by means of the dominant singularity approximation. We show that obtaining these tail probabilities can be a complex task, and that the obtained tail probabilities are not necessarily exponential (as in most ‘traditional’ queueing systems). Further, we show the impact and significance of the various system parameters on the type of tail behavior. Finally, we compare our approximation results with simulations.     

Sharp large deviations for sums of bounded from above random variables

We show large deviation expansions for sums of independent and bounded from above random variables. Our moderate deviation expansions are similar to those of Cram′er(1938), Bahadur and Ranga Rao(1960), and Sakhanenko(1991). In particular, our results extend Talagrand's inequality from bounded random variables to random variables having finite(2 + δ)-th moments, where δ∈(0, 1]. As a consequence,we obtain an improvement of Hoeffding's inequality. Applications to linear regression, self-normalized large deviations and t-statistic are also discussed.     

Bounds for some general sums of random variables

Rogers and Shi (1995) have used the technique of conditional expectations to derive approximations for the distribution of a sum of lognormals. In this paper we extend their results to more general sums of random variables. In particular we study sums of functions of dependent random variables that are multivariate normally distributed and also derive results for sums of functions of dependent random variables from the additive exponential dispersion family. The usefulness of our results for practical applications is also discussed.     

Multivariate Discrete Distributions with a Product-Type Dependence

A discrete multivariate probability distribution for dependent random variables, which contains the Poisson and Geometric conditionals distributions as particular cases, is characterized by means of conditional expectations of arbitrary one-to-one functions. Independence of the random variables is also characterized in terms of these conditional expectations. For certain exchangeable and partially exchangeable random variables with a joint distribution of this form it is shown that maximum likelihood estimates coincide with the simple method of moments estimates, suggesting that these models offer a pragmatic way to analyze certain dependent data.     

保费收取过程是随机过程的双险种风险模型

研究保费收取过程是一个随机过程的双险种风险模型,得出了Lundberg上界、最终破产概率、不破产所满足的微积分方程、索赔服从指数分布的不破产概率、有限时间不破产所满足的微积分方程.     

基于对称随机变量构造的反例

应用相关文献中对称随机变量分布函数的充要条件,阐明连续型对称随机变量概率密度的偶函数特点,以及对称随机变量的不相关性,构造一些教学反例.     

Transforms of pseudo-Boolean random variables

As in earlier works, we consider {0,1}ⁿ as a sample space with a probability measure on it, thus making pseudo-Boolean functions into random variables. Under the assumption that the coordinate random variables are independent, we show it is very easy to give an orthonormal basis for the space of pseudo-Boolean random variables of degree at most k. We use this orthonormal basis to find the transform of a given pseudo-Boolean random variable and to answer various least squares minimization questions.     

Large‐deviations/thermodynamic approach to percolation on the complete graph

We present a large‐deviations/thermodynamic approach to the classic problem of percolation on the complete graph. Specifically, we determine the large‐deviation rate function for the probability that the giant component occupies a fixed fraction of the graph while all other components are “small.” One consequence is an immediate derivation of the “cavity” formula for the fraction of vertices in the giant component. As a byproduct of our analysis we compute the large‐deviation rate functions for the probability of the event that the random graph is connected, the event that it contains no cycles and the event that it contains only small components. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Poset limits and exchangeable random posets

We develop a theory of limits of finite posets in close analogy to the recent theory of graph limits. In particular, we study representations of the limits by functions of two variables on a probability space, and connections to exchangeable random infinite posets.