Probability Distributions Of Random Variables Research Articles

Network engineering for agile belief network models

The construction of a large, complex belief network model, like any major system development effort, requires a structured process to manage system design and development. This paper describes a belief network engineering process based on the spiral system lifecycle model. The problem of specifying numerical probability distributions for random variables in a belief network is best treated not in isolation, but within the broader context of the system development effort as a whole. Because structural assumptions determine which numerical probabilities or parameter values need to be specified, there is an interaction between specification of structure and parameters. Evaluation of successive prototypes serves to refine system requirements, ensure that modeling and elicitation effort are focused productively, and prioritize directions of enhancement and improvement for future prototypes. Explicit representation of semantic information associated with probability assessments facilitates tracing of the rationale for modeling decisions, as well as supporting maintenance and enhancement of the knowledge base.

Phase Transition in Ferromagnetic Transverse Spin-1/2 Ising Multilayers

The critical behavior of ferromagnetic transverse spin-1/2 Ising multilayers consisting of two different ferromagnetic materials A and B and presenting a sandwich structure in the form ABA, is examined using the effective field theory. This method takes into consideration the fluctuations of the effective field, and is based on a probability distribution of random variables which correctly accounts for all the single site kinematic relations. The critical temperature of the system is studied as a function of the thickness of the constituents in a multilayer and of exchange coupling in each material. A critical value of the interface exchange coupling above which the interlace magnetism appears is found. The effects of the transverse field, the interface exchange coupling, and surface exchange coupling on the cptical behavior of the system are also investigated.

The ferromagnetic transverse spin-1/2 Ising superlattice

The critical behavior of the ferromagnetic transverse spin-1/2 Ising superlattice on a simple cubic structure consisting of two different ferromagnet materials is examined using the effective field theory. This method takes into consideration the fluctuations of the effective field, and is based on a probability distribution of random variables which correctly accounts for all the single site kinematic relations. The critical temperature of the system is studied as a function of the thickness of the constituents in a unit cell and of exchange coupling in each material. A critical value of the interface exchange coupling above which the interface magnetism appears is found. The effects of the transverse field and the interface exchange coupling on the critical behavior of the system are also investigated.

Phase Transitions of Ferromagnetic Ising Films withAmorphous Surfaces

The critical behavior of a ferromagnetic Ising film with amorphoussurfaces is studied within the framework of the effective fieldtheory. The method takes into consideration the fluctuation of theeffective field, and is based on a probability distribution of randomvariables which correctly accounts for all the single site kinematicrelations. The dependence of the critical temperature on exchangeinteraction strength ratio, film thickness, and amorphization ispresented. It is found that an order–disorder magnetictransition occurs by varying the thickness of the film. Such a resultis in agreement with experiments performed recently onFe-films.

The Random Transverse Ising Model with Spin-1/2, Spin-1 and Mixed Spins

Using the effective field theory (EFT) which takes into consideration the fluctuations of the effective field and which is based on a probability distribution of random variables which accounts for all the single kinematic relations, we have studied the transverse random field Ising model (TRIM) with spin-1/2 and spin-1 and the transverse random field mixed Ising model (TRMIM) with both spin-1/2 and spin-1. It is shown that the simplest approximation gives satisfactory results for the phase diagrams when compared with other methods.

Maximum trimmed likelihood estimators: a unified approach, examples, and algorithms

The likelihood principle is one of the most important concepts in Statistics. Among other things, it is used to obtain point estimators for the parameters of probability distributions of random variables by maximizing the likelihood function. The resulting maximum likelihood estimators usually have desirable properties such as consistency and efficiency. However, these estimators are often not robust as it is known that there is usually a trade-off between robustness and efficiency; the more robust an estimator is the less efficient it may be when the data come from a Gaussian distribution. In this paper we investigate how the estimators change when the likelihood function is replaced by a trimmed version of it. The idea here is to trim the likelihood function rather than directly trim the data. Because the likelihood is scalar-valued, it is always possible to order and trim univariate as well as multivariate observations according to their contributions to the likelihood function. The degree of trimming depends on some parameters to be specified by the analyst. We show how this trimmed likelihood principle produces many of the existing estimators (e.g., maximum likelihood, least squares, least trimmed squares, least median of squares, and minimum volume ellipsoid estimators) as special cases. Since the resulting estimators may be very robust, they can be used for example for outliers detection. In some cases the estimators can be obtained in closed form. In other cases they may require numerical solutions. In cases where the estimators cannot be obtained in closed forms, we provide several algorithms for computing the estimates. The method and the algorithms are illustrated by several examples of both discrete and continuous distributions. © 1997 Elsevier Science B.V.

Reliability analysis of emergency and standby power systems

Different facilities have varying requirements for reliability of electric power supply. The reliability indices of an actual physical system can be estimated by collecting data on the occurrence of failures and the durations of repair. The Monte Carlo method mimics the failure and repair history of the components and the system by using the probability distributions of the component state durations. Statistics are then collected and indices estimated using statistical inference. There are two basic approaches for Monte Carlo simulation: (1) sequential simulation and (2) random sampling. The sequential simulation proceeds by generating a sequence of events using random numbers and probability distributions of random variables representing component state durations. In random sampling, states are drawn based on the probability distributions of component states and random numbers. Further, there are two methods for representing the passage of time in sequential simulation: (1) the fixed interval method, also called synchronous timing, and (2) the next event or asynchronous timing method. In the fixed interval method, time is advanced in steps of fixed length and the system state is updated. In the next event method, time is advanced to the occurrence of the next event. In actual implementations, it is likely that combinations of the timing controls may be used. The sampling method is generally faster than the sequential technique, but is suitable when component failures and repairs are independent. This article presents the sequential method for reliability analysis.

A new look at the response surface approach for reliability analysis

Closed-form mechanical models to predict the behaviour of complex structural systems often are unavailable. Although reliability analysis of such systems can be carried out by Monte Carlo simulations, the large number of structural analyses required results in prohibitively high computational costs. By using polynomial approximations of actual limit states in the reliability analysis, the number of analyses required can be minimized. Such approximations are referred to as Response Surfaces. This paper briefly describes the response surface methodology and critically evaluates existing approaches for choosing the experimental points at which the structural analyses must be performed. Methods are investigated to incorporate information on probability distributions of random variables in selecting the experimental points and to ensure that the response surface fits the actual limit state in the region of maximum likelihood. A criterion for reduction in the number of experiments after the first iteration is suggested. Two numerical examples show the application of the approach.

Throughput Analysis of Multiple-Bus Multiprocessor Systems with Simultaneous Possession of Common Resources

To answer the questions for the performance issues of multiple-bus multiprocessor systems under general conditions, a discrete-event simulation model has been developed. This model has been analyzed for different types of probability distributions for random variables A = processing times and B = memory access times for processors. For variables A and B, the following types of distributions are employed: exponential, Erlang, hypoexponential, hyperexponential, normal and constant. Simulation experiments show that the coefficient of variation of B, CB has a significant effect on the system performance, especially for CB>1, but the effect of CA is negligible if A is either hypo- or hyperexponentially distributed. However, this is not the case for a normal distribution. Processing efficiency, memory utilization and the percent of queued processors are some of the performance metrics used in our simulation model. We simulated several multiprocessor configurations and reach the conclusion that if the number of buses is equal to half the minimum value of the number of processors and the number of memory modules, the corresponding bus-deficient system behaves as a bus-sufficient system. Clearly, this is a substantial saving for the required number of buses without degrading the system performance.

確率計画問題における信頼域によるアプローチ

We propose a minirnax model with a quadratic recourse. In stochastic linear programming models, a decision maker has been assumed to know the probability distribution of random variables. Here we consider the case that the parameters of distribution are unknown. We impose the restrictions on the unknown parameters from the view point of a confidence region, and then seek a minimax solution that minimizes the worst case of the parameters. This model reflects the situation minimizing the maximal possible damage. Especially, the independent normal distribution model is discussed in detail. The analysis for a sufficiently large sample size and a numerical result are given.

On some peculiarities of quantum mechanics

General regularities related toLagrangian andHamiltonian equations are revealed. Probability distributions for functions ofHamiltonian random variables are considered. It is shown that all probability distributions of this kind are fully determined by the probability distributions for the random variables satisfying the corresponding Lagrangian equations. Some formulas related tocanonically conjugate operators are given. The similarity of these formulas to those related to Hamiltonian random variables is demonstrated. The “quantum approach” to the treatment of Hamiltonian random variables is discussed, and the origin of some peculiarities related to this approach is elucidated; it is explained, in particular, why it is impossible to form the joint probability density for canonically conjugate random variables when using this approach. The peculiarities revealed prove to be common for any objects possessing Hamiltonian random variables, irrespective of the nature of the objects, and coincide, therefore, with those in quantum mechanics. The existence of joint probability distributions for canonically conjugate random variables in the general case is demonstrated through the calculation of the corresponding joint mathematical expectations in an illustrative example. This proves, in particular, that joint probability distributions for canonically conjugate coordinates and momenta exist indeed in the case of mechanical microsystems. The results obtained prove once again that the pecularities of quantum mechanics are not related to the specificity of the measurements of physical quantities for microsystems.

Calculation of Multidisturber Crosstalk Probabilities-Application to Subscriber-Loop Gain

The possibility that excessive crosstalk limits the application of gain to long voice-frequency loops was studied. This problem was examined by means of a probability model incorporating data on such random variables as activity coincidence of disturbing and disturbed subscribers, disturber volume level, coupling path loss, listener acuity, and various noise sources. A new approach was used with regard to the probability distributions of random variables: the probability distributions were modelled in detail for the calculation of the necessary functions, sums, and functions of sums of random variables. Results showed that gain of 6 dB or less is acceptable, and 9 dB is unjustifiably excessive. The approach used provides the most accurate calculation possible with available data, and is anticipated to be convenient not only for similar crosstalk evaluations but also for other nongaussian probability problems.

Probability Distributions of Random Variables Associated with a Structure of the Sample Space of Sociometric Investigations

In this paper, we consider a disjoint decomposition, at three levels, of the total sample space for $n$-person, one-dimensional sociometric investigations. This results in a structure particularly suited to determination of the probability distributions of a large class of sociometric variables. Systematic methods for obtaining these distributions are presented and illustrated by two examples; while the first is trivial, the second produces a previously unknown result. It should be remarked that the methods developed here have application in the theory of communication networks and, indeed, in the study of any network situations which may be represented by either of the two models employed in the paper.

The number of locally restricted directed graphs

1. Preliminaries. We shall be concerned with finite graphs of t directed lines on n points, or nodes. The lines are joinsfrom one point to another. To each point, Pi, is associated a non-negative number pair (ri, si), where ri is the number of lines from Pi, and si is the number of lines to Pi. These number pairs are considered to be fixed local restrictions. Obviously, ri= Ej sj, =t. The vectors r and s, with elements ri and si, respectively, are two n-part non-negative integer partitions of t, with vanishing parts permitted. In general, it is possible to construct many n-node graphs satisfying a specified set of local restrictions. The main result of this paper gives the number of distinct graphs of this kind. The enumeration is helpful in studying communication networks in which the ith element has ri receivers and si senders; here the pertinent question is whether such graphs exist. In the isomorphic problem in group organization theory, where the ith individual in the group gives information or orders or choices to ri others and receives from si others, the exact number of graphs is needed in order to construct the probability distributions for random variables defined over the graphs. All of these activities, like the graphs, are irreflexive and take values on a twoelement Boolean algebra. Previous work on numbers of graphs has been devoted exclusively to counts of graphs topologically distinct under permutations of the points which preserve joins. Thus, G. Polya [9] gave explicitly the numbers of trees and rooted trees on n points, R. Otter [8] gave simpler purely combinatorial methods for counting trees and rooted trees, and F. Harary and G. E. Uhlenbeck [6] gave the numbers of free pure and mixed Husimi trees. Also, F. Harary indicates in an abstract [5] that Polya's method of generating functions may be extended to give the numbers of ordinary and of directed graphs on n points and t lines. R. L. Davis [3] defines and gives a method for counting the numbers of various subclasses of directed graphs on n points. The problem we face is essentially different from all of these and requires quite different methods of enumeration. We shall make use of certain bipartitional functions due to Cayley