Moments Of Random Variables Research Articles

Reliability Analysis for Stiffened Plate on the Maximum Entropy Method

Stiffened plate is widely used in vessel structure because of its high bearing capacity and low weight so the research of failure probability for stiffened plate under explosion load has important engineering meaning. Stiffened plate under near-field explosion is taken as research subject, dynamite density and yield stress of plate are selected as random variables, the original values of one hundred groups of random variables are gotten through the random number generation program, and the moments of random variables are obtained. Based on failure criterion of displacement ductility, the performance function of structure is established, probability density function of performance function is fitted using maximum entropy method then the failure probability of stiffened plate structure is obtained. So as to solve the problem of calculate failure probability when the sample size is small and the probability density function is unknown.

Saddlepoint approximation for moments of random variables

In this paper, we introduce a saddlepoint approximation method for higher-order moments like E(S − a) + , a>0, where the random variable S in these expectations could be a single random variable as well as the average or sum of some i.i.d random variables, and a > 0 is a constant. Numerical results are given to show the accuracy of this approximation method.

Robust Solutions of Optimization Problems Affected by Uncertain Probabilities

In this paper we focus on robust linear optimization problems with uncertainty regions defined by o-divergences (for example, chi-squared, Hellinger, Kullback-Leibler). We show how uncertainty regions based on o-divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector. Such problems frequently occur in, for example, optimization problems in inventory control or finance that involve terms containing moments of random variables, expected utility, etc. We show that the robust counterpart of a linear optimization problem with o-divergence uncertainty is tractable for most of the choices of o typically considered in the literature. We extend the results to problems that are nonlinear in the optimization variables. Several applications, including an asset pricing example and a numerical multi-item newsvendor example, illustrate the relevance of the proposed approach.

A Modified High-Order Moment Method Based on the Normal Transformation Polynomial

Normal transformation technique is often used in practical probabilistic analysis in structural or civil engineering especially when multivariate random variables with the probabilistic characteristics expressed using only statistical moments are involved. In this paper, a modified high-order moment method(MHOM) is given based on the polynomial coefficients of a third-order normal transformation polynomial (NTP) using the first four central moments of random variables having unknown distributions. The present high-order moment method is introduced into several typical test problems having unknown distributions are available. Since it needs neither the computation of derivatives nor iteration, and since it is unnecessary to know the probability distribution of the basic random variables, the present method should be practical in actual reliability problems. Applications to several typical examples have helped to elucidate the successful working of the present MHOM.

Plate response variability due to triple random parameters

In this paper, we aim to propose a stochastic finite element formulation having a capability of considering triple random parameters, i.e. elastic modulus, Poisson’s ratio and thickness of plate, simultaneously in the evaluation of response variability of plates. The effects of cross-correlations between the three random parameters on the response variability are also observed. The generalized auto- and cross-correlation functions that are needed in the evaluation of response covariance are derived based on the general expression for the n-th moment of random variables. In order to demonstrate the adequacy of the proposed scheme, we employ a square plate with various boundary conditions as an example, and the results are compared with those given in the literature as well as with the results of Monte Carlo analysis.

Set partitions and moments of random variables

It is well known that the sequence of Bell numbers ( B n ) n ⩾ 0 ( B n being the number of partitions of the set [ n ] ) is the sequence of moments of a mean 1 Poisson random variable τ (a fact expressed in the Dobiński formula), and the shifted sequence ( B n + 1 ) n ⩾ 0 is the sequence of moments of 1 + τ . In this paper, we generalize these results by showing that both ( B n 〈 m 〉 ) n ⩾ 0 and ( B n + 1 〈 m 〉 ) n ⩾ 0 (where B n 〈 m 〉 is the number of m-partitions of [ n ] , as they are defined in the paper) are moment sequences of certain random variables. Moreover, such sequences also are sequences of falling factorial moments of related random variables. Similar results are obtained when B n 〈 m 〉 is replaced by the number of ordered m-partitions of [ n ] . In all cases, the respective random variables are constructed from sequences of independent standard Poisson processes.

Robust Solutions of Optimization Problems Affected by Uncertain Probabilities

In this paper we focus on robust linear optimization problems with uncertainty regions defined by ϕ-divergences (for example, chi-squared, Hellinger, Kullback–Leibler). We show how uncertainty regions based on ϕ-divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector. Such problems frequently occur in, for example, optimization problems in inventory control or finance that involve terms containing moments of random variables, expected utility, etc. We show that the robust counterpart of a linear optimization problem with ϕ-divergence uncertainty is tractable for most of the choices of ϕ typically considered in the literature. We extend the results to problems that are nonlinear in the optimization variables. Several applications, including an asset pricing example and a numerical multi-item newsvendor example, illustrate the relevance of the proposed approach.This paper was accepted by Gérard P. Cachon, optimization.

Advanced method to estimate reliability-based sensitivity of mechanical components with strongly nonlinear performance function

Based on the random perturbation technique for reliability sensitivity design, some realistic reliability-based sensitivity issues are discussed, some of which have a structure of high nonlinear performance functions. Combining the related theories of the moment method of the reliability analysis, the matrix differential, and the Kronecker algebra, the reliability-based sensitivity method based on the perturbation method is modified if the first four moments of random variables are given. Meanwhile, a reliability-based sensitivity computation method is proposed. Some examples are used to show that using this method can effectively improve the accuracy of the reliability-based sensitivity computation and offer a reliable theoretic basis in engineering.

A population based approach to model the lifetime and energy distribution in battery constrained wireless sensor networks

The residual power levels of the nodes in a wireless sensor network determine its important performance metrics like the network lifetime, coverage, and connectivity. In this paper, we present a general framework to model the availability of power at sensor nodes as a function of time, based on models for population dynamics in biological studies. Models are developed for sensors with and without battery recharging and expressions are derived for the network lifetime as well as the distribution and moments of random variables describing the number of sensors with different levels of residual energy as a function of time. The model is also extended to the case where new sensors are periodically added to the network to substitute older sensors that have expended their energy. Finally, the effect of the packet arrival rates and a sensor's geographical location are modeled. Simulation results to verify the accuracy of the proposed models are presented.

Maximal inequalities for some dependent sequences and their applications

In this paper, we obtain the maximal inequalities for linear process, φ -mixing sequence and linearly negative quadrant dependent (LNQD) sequence when the r th moments of random variables are finite for r > 2 . Applying these maximal inequalities above, we get the Hájek–Rényi-type inequality, strong law of large numbers, strong growth rate and integrability of supremum for these three sequences.

Parameter estimation for generalized Dirichlet distributions from the sample estimates of the first and the second moments of random variables

Generalized Dirichlet distributions have a more flexible covariance structure than Dirichlet distributions, and the computation for the moments of a generalized Dirichlet distribution is still tractable. For situations under which Dirichlet distributions are inappropriate for data analysis, generalized Dirichlet distributions will generally be an applicable alternative. When the expected values and the covariance matrix of random variables can be estimated from available data, this study introduces ways to estimate the parameters of a generalized Dirichlet distribution for analyzing compositional data. Under the assumption that the sample mean of every variable must be considered for parameter estimation, we present methods for choosing the statistics from a sample covariance matrix to construct a generalized Dirichlet distribution. Some rules for removing inappropriate statistics from a sample covariance matrix to speed up the estimation process are also established. An example for Taiwan’s car market is introduced to demonstrate the applicability of the parameter estimation methods.

Effects of random material and geometrical properties on structural safety of steel-concrete composite systems

A stochastic finite element-based algorithm for probabilistic analysis of structural systems made of composite sections with random material and geometrical properties under earthquake forces is proposed in this paper. Uncertainties in the structural parameters can be taken into account in this algorithm. For the perturbation-based stochastic finite element method, only the first two moments of random variables need to be known, and is numerically much more efficient and feasible than simulation techniques. The efficiency and accuracy of the proposed algorithm are validated by comparison with results of Monte Carlo simulation method. A summary of stiffness matrix formulation and perturbation-based stochastic finite element dynamic analysis formulation of structural systems made of composite sections is given. These are followed by suitable numerical examples, which indicate that employment of such a dynamic stochastic finite element method leads to significant economical, efficient and accurate solutions for the dynamic analysis of composite structures with stochastic parameters under earthquake forces. Copyright © 2010 John Wiley & Sons, Ltd.

An analytical approach for performance evaluation and optimization of a two-stage production-distribution system

This paper addresses the performance evaluation and optimization of a production-distribution system composed of a warehouse supplied by a manufacturing plant with finite capacity. Customer orders arrive randomly with random order size. Transportation time from plant to warehouse is fixed. We propose an analytical approach that uses only the first two moments of random variables to evaluate the inventory cost and demand fill rate. A gradient-based method is then proposed to minimize the average total inventory cost subject to fill rate requirement. Numerical comparisons with simulation show that the analytical approach is very efficient.

Perturbation Based Stochastic Finite Element Analysis of the Structural Systems with Composite Sections under Earthquake Forces

This paper demonstrates an application of the perturbation based stochastic finite element method (SFEM) for predicting the performance of structural systems made of composite sections with random material properties. The composite member consists of materials in contact each of which can surround a finite number of inclusions. The perturbation based stochastic finite element analysis can provide probabilistic behavior of a structure, only the first two moments of random variables need to be known, and should therefore be suitable as an alternative to Monte Carlo simulation (MCS) for realizing structural analysis. A summary of stiffness matrix formulation of composite systems and perturbation based stochastic finite element dynamic analysis formulation of structural systems made of composite sections is given. Two numerical examples are presented to illustrate the method. During stochastic analysis, displacements and sectional forces of composite systems are obtained from perturbation and Monte Carlo methods by changing elastic modulus as random variable. The results imply that perturbation based SFEM method gives close results to MCS method and it can be used instead of MCS method, especially, if computational cost is taken into consideration.

Moment representation of Bernoulli polynomial, Euler polynomial and Gegenbauer polynomials

The Hermite polynomials can be represented to the moments of normal distribution by the work of Withers [2000. A simple expression for the multivariate Hermite polynomials. Statist. Probab. Lett. 47, 165–169]. This paper generally shows certain combinatorial polynomials and orthogonal polynomials are also the moments of random variables, such as Bernoulli polynomials, Euler polynomials, Gegenbauer polynomials.

Estimating Load and Resistance Factors without Using Distributions of Random Variables

In this paper, a new method for estimating the load and resistance factors using the first three moments of random variables is proposed. Unlike the currently used method, the load and resistance factors can be determined without using distributions of random variables, and the present method needs neither the iterative computation of derivatives nor any design points. The present method can effectively reflect the characteristics of the skewness of random variables and the performance function, and generally provide much more accurate results than the second moment method. Thus, the present method should be convenient and more effective in estimating the load and resistance factors in practical engineering. Numerical examples are presented to demonstrate the advantages of the present method.

On calculation of moments of ladder heights

Let X,X 1,X 2, … be independent identically distributed random variables, F(x) = P{X < x}, S 0 = 0, and S n =Σ =1 X i . We consider the random variables, ladder heights Z + and Z − that are respectively the first positive sum and the first negative sum in the random walk {S n }, n = 0, 1, 2, …. We calculate the first three (four in the case EX = 0) moments of random variables Z + and Z − in the qualitatively different cases EX > 0, EX < 0, and EX = 0.

Added effect of uncertain geometrical parameter on the response varibility of Mindlin plate

In case of Mindlin plate, not only the bending deformation but also the shear behavior is allowed. While the bending and shear stiffness are given in the same order in terms of elastic modulus, they are in different order in case of plate thickness. Accordingly, bending and shear contributions have to be dealt with independently if the stochastic finite element analysis is performed on the Mindlin plate taking into account of the uncertain plate thickness. In this study, a formulation is suggested to give the response variability of Mindlin plate taking into account of the uncertainties in elastic modulus as well as in the thickness of plate, a geometrical parameter, and their correlation. The cubic function of thickness and the correlation between elastic modulus and thickness are incorporated into the formulation by means of the modified auto- and cross-correlation functions, which are constructed based on the general formula for n-th joint moment of random variables. To demonstrate the adequacy of the proposed formulation, a plate with various boundary conditions is taken as an example and the results are compared with those obtained by means of classical Monte Carlo simulation.

A Grüss-type inequality and its applications

We give a Gruss-type inequality which is a refinement of a result due to Dragomir and Agarwal. We also give its applications for the moments of random variables, guessing mappings, and Ozeki's inequality.

Use of the bulk harmonic fall velocity in mechanical calculations at two levels of input data

Many current problems of ecological mechanics must be solved with incomplete information about the motion of large aggregates of liquid and solid particles in liquid and gas flows [1–5]. These problems require the averaging of physical fields [6], calculation of the moments of random variables [7], and experimental determination of the bulk characteristics of particles [1, 2, 5, 8] and their use in mechanical calculations. One of these characteristics is the bulk harmonic (or mean harmonic) fall velocity of particles. As will be shown below, this quantity can be directly determined from the curve of sediment-mass accumulation for small concentrations of particles in a stationary liquid. The comparison of two expressions for the bulk time of the sedimentation of particles indicates that