Moments Of Random Variables Research Articles

A Coordinate-Free Approach to the Method of Moments in the Theory of Multidimensional Stochastic Systems

A coordinate-free approach to the theory of the distribution moments of random variables was considered. The main relations for the moments were obtained using the theory of symmetric polylinear forms. Problems of polynomial approximation of the distributions of the state vector of multidimensional stochastic systems were discussed.

Uncertainty Analysis by Point Estimate Methods Incorporating Marginal Distributions

The model performance of an engineering system is affected by many variables subject to uncer­ tainty. Point estimate (PE) methods are practical tools to assess the uncertainty features of a model involving multivariate stochastic parameters. Two PE methods have been developed for engineering applications. One is Rosenblueth's PE method, which preserves the first three moments of random variables and the other is Hare's PE method, which reduces the computations of Rosenblueth's method but only as appropriate for application to random variables with normal distributions. In this study, two algorithms are proposed to encompass the advantages of the two PE methods: computational practicality and the handling of mixture distributions. Through a numerical experiment, the proposed methods yielded more accurate estimations than those of Rosenblueth's method with about the same amount of computation as Hare's method. The two proposed methods were also applied to estimate statistical moments of a pier scouring model output to demonstrate their performance in an engineering application.

On the moments of random variables uniformly distributed over a polytope

Suppose X = (X1, X2, …, Xn) is a random vector uniformly distributed over a polytope. In this note, the author derives a formula for , (the expected value of ), in terms of the extreme points of the polytope.

Analysis methods for optical heterodyne DPSK receivers corrupted by laser phase noise

Two methods are presented for analyzing the effects of phase noise on the performance of an optical heterodyne binary differential-(DPSK) system. The first method utilizes a perturbation solution for filtered phase noise. By comparing the results of this analysis with simulated results, it is shown that the perturbation solution is accurate for laser linewidths up to at least 10% of the bit rate. Using this analysis, the accuracy of the widely used approximation, whereby the effects of filtering on the magnitude of the phase-noise corrupted signal are neglected, is verified. The author's second method is based on moments of random variables. As the level of phase noise in a practical DPSK system must be small, an improved formulation for the moments of the filtered phase noise is derived. It removes the major cause of this numerical instability. A maximum-entropy probability density function estimation technique is applied to the problem of analyzing the performance of a DPSK receiver. By comparing results with those obtained using the perturbation analysis, it is found that the moment-based method is effectively limited to relatively large error probabilities.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A note on a method to compute the asymptotic Distribution of the sample second order moments of dynamic linear normal variables

This paper describes a method for the computation of the asymptotic distribution of any statistic or estimator which is a continuous function of sample second order moments of random variables having a dynamic linear normal distribution of the form : Zt = IIt-1 + et where et iN(0, σ) The method uses a general theorem due to Hannan [1970] and simplifies it in decomposing II in Jordan canonical form.

Stochastic Finite Element Analysis of Buckling of Frame Structure with Uncertain Initial Stresses

When members having uncertainty in length are assembled into a frame structure, uncertain initial stresses are generated in it. This may influence the buckling load which is related with structural safety and reliability. Stochastic finite element method on the basis of the second order perturbation technique is developed in this class of problem and the formation for two dimensional frame structure is presented. The rates of change of eigenvalue for plural random variables are efficiently computed based on the single solution of the governing equation. It is demonstrated in numerical examples that the expectation and variance of buckling load are successfully evaluated for given moments of random variables.

Real‐time forecasting with a conceptual hydrologic model: 1. Analysis of uncertainty

The optimal control of watershed systems requires accurate real‐time short‐term forecasts of river flows. For the first time, this paper formulates a large, nonlinear conceptual model (the National Weather Service catchment model) in a mode amenable to analysis of uncertainty and the utilization of real‐time information (measurements, forecasts, guesses) to update system states and improve streamflow predictions. The proposed methodology is based on the state space formulation of the equations describing the hydrologic model and the assumption of sources of uncertainty in the data and in the model structure. The first two moments of random variables are estimated in a computationally efficient way using on‐line linear estimation techniques. Linearization of functional relationships is performed with the uncommon but powerful multiple‐input describing function technique for the most strongly nonlinear responses and Taylor expansion for the rest. The linear feedback rule developed is based on the Kalman filter.

Comments on uniqueness of statistics derived from moments of random variables

Electrical engineers and physicists are often confronted with the problem of finding a probability density of a random variable (e.g., a noisy signal) from its moments. Some sources of apparent confusion in deciding whether the distribution is uniquely determined are discussed.

Bounds on Conditional Moments of Weibull and other Monotone Failure Rate Families

Abstract Bounds are obtained for conditional moments of random variables X, such as E(X|X > a), and of order statistics from members of monotone failure rate families of distributions. Numerical bounds for Weibull and other families are calculated and compared. Results classifying some functions of random variables as having monotone failure rate distributions are obtained, as are characterizations of absolutely continuous distributions with cdf's of the form 1 – exp[–H(x)], x > 0.