Showing papers on "Probability distribution published in 1968"

Approximating discrete probability distributions with dependence trees

Abstract: A method is presented to approximate optimally an n -dimensional discrete probability distribution by a product of second-order distributions, or the distribution of the first-order tree dependence. The problem is to find an optimum set of n - 1 first order dependence relationship among the n variables. It is shown that the procedure derived in this paper yields an approximation of a minimum difference in information. It is further shown that when this procedure is applied to empirical observations from an unknown distribution of tree dependence, the procedure is the maximum-likelihood estimate of the distribution.

Games with Incomplete Information Played by 'Bayesian' Players, Part III. The Basic Probability Distribution of the Game

Abstract: Parts I and II of this paper have described a new theory for the analysis of games with incomplete information. Two cases have been distinguished: consistent games in which there exists some basic probability distribution from which the players' subjective probability distributions can be derived as conditional probability distributions; and inconsistent games in which no such basic probability distribution exists. Part III will now show that in consistent games, where a basic probability distribution exists, it is essentially unique. It will also be argued that, in the absence of special reasons to the contrary, one should try to analyze any given game situation with incomplete information in terms of a consistent-game model. However, it will be shown that our theory can be extended also to inconsistent games, in case the situation does require the use of an inconsistent-game model.

Metric: A Multi-Echelon Technique for Recoverable Item Control

Abstract: Metric is a mathematical model of a base-depot supply system in which item demand is compound Poisson with a mean value estimated by a Bayesian procedure. When a unit fails at base level there is a probability r that it can be repaired at the base according to an arbitrary probability distribution of repair time, and a probability 1 − r that it must be returned to the depot for repair according to some other arbitrary distribution. In the latter case the base levies a resupply request on depot. No lateral resupply between bases is considered in the model. For high-cost, low-demand items the appropriate policy is (s − 1, s), which means that items are not batched for repair or resupply requests. This problem has a simple analytic solution that is a function of the mean repair times rather than the repair time distributions. A practical and efficient computer program has been designed to show the cost-effectiveness tradeoff for a large group of recoverable items. In addition to minimizing expected backorder...

The Consensus of Subjective Probability Distributions

Abstract: “‘But we can't agree whether A or B is correct,' he concluded, ‘and so we're collecting expert opinions, weighting them appropriately, and programming WESCAC to arbitrate the whole question.’” (John Barth, Giles Goat-Boy, p. 664.) In the Bayesian framework, quantified judgments about uncertainty are an indispensable input to methods of statistical inference and decision. If a decision maker has little knowledge with regard to the parameters of interest, he may decide to consult a number of experts and obtain their quantified judgments in the form of subjective probability distributions. If this is the case, the decision maker must somehow combine the distributions assessed by the experts and form a single distribution to be used as an input to a formal Bayesian analysis. Several methods for combining the distributions are suggested, some involving mathematical formulae and some involving feedback and/or group discussion. These methods are compared under certain assumptions regarding the form of the distri...

The theory of probability

Abstract: In the 19th century, both the ideas and methods of the theory of probability, developed since the second half of the 17th century, received new incentives for further progress. These stimuli, highly differing one from another in their essence, were connected with the development of the natural sciences, with practical requirements of society and, also, with the formulation of purely mathematical problems.

Discrete Stochastic Programming

Abstract: A method is presented for solving linear programming problems where (any number of) the functional, restraint, and input-output coefficients are subject to discrete; probability distributions. The objective function is formulated in terms of variance and/or expectation. The procedure involves the simultaneous generation of all (mutually exclusive) possible outcomes and hence the transference of all variability into the objective function of a very much enlarged linear program.

The multiple scattering of waves in irregular media

Abstract: When a wave passes through a large thickness of a non-absorbing medium containing weak random irregularities of refractive index, large amplitude and phase fluctuations of the wave field can develop. The probability distributions of these fluctuations are important, since they may be readily observed and from them can be found the mean square amplitudes of the fluctuations. This paper shows how to calculate these distributions and also the ‘ angular power spectrum ’ for an assembly of media which are statistically stationary with respect to variations in time, and in space for directions perpendicular to the wave normal of the incident wave. The scattered field at a given point is resolved into two components in phase and in quadrature with the residual unscattered wave at that point. The assembly averages of the powers in these two components, and of their correlation coefficient are found, and a set of three integro-differential equations is constructed which show how these three quantities vary as the medium is traversed. The probability distributions of amplitude and phase of the wave field at any point in the medium are functions of these three quantities which are found by integrating the equations through the medium. An essential feature of these equations is that they include waves which have been scattered several or m any times (multiple scatter). The equations are solved analytically for some particular cases. Solutions for the general case have been obtained numerically and are presented, together with the corresponding probability distributions of the field fluctuations and their average values.

Interprétation de la diffusion centrale des rayons X par les systèmes poreux. I

Abstract: A porous system may be characterized by using two statistical distributions of chord lengths: \varphi(l) (particle chords) and f(m) (pore chords). Calculations are presented giving a general relationship between the shape of small angle scattering and the distribution of segment lengths limited by particle and pore boundaries. This development represents a generalization of Porod's method. By means of an approximation, this general expression is simplified and can be applied in many cases. The properties of distributions \varphi(l) [or f(m)] are analysed and it is shown that the condition \varphi(0) = 0 (or f(0) = 0] means that particles (or pores) do not possess any sharp edges. The presence or absence of sharp edges allows the separation of small angle scattering curves into two characteristic forms. The functions \varphi(l) and f(m) corresponding to several simple geometrical forms are analysed.

Random matrices and information theory

Abstract: A general method is proposed for assigning a probability distribution to a random matrix Its principle is that the amount of information contained in the probability distribution should not exceed the minimum amount needed to satisfy relevant properties of the matrix As examples, classical random matrices are recovered, and the case of a given density of eigenvalues is treated

Headstart Strategies for Combating Congestion

Abstract: When a traveler, or other user of a congested system, wishes to be reasonably sure of reaching a destination, or finishing a task, on time he must start early enough to compensate for random delays. This paper investigates the headstart decision policy arising if costs are attached both to lateness and to an otherwise undesirable early departure. The model encompasses a case in which the probability distribution of congestion delays is unknown, and must be inferred from previous experience.

On the behaviour of the characteristic function of a probability distribution in the neighbourhood of the origin

Abstract: Let X be a real valued random variable with probability measure P and distribution function F. It will be convenient to take F as the intermediate distribution function defined by . In mathematical analysis it is a little more convenient to use this function rather than , which arise more naturally in probability theory. In all cases we shall consider With this definition, if the distribution function of X is F(x), then the distribution function of −X is 1−F(−x). The distribution of X is symmetrical about 0 if F(x) = 1 − F(−x).

Response of the Bilinear Hysteretic System to Stationary Random Excitation

Abstract: Time‐average statistics of the response of the bilinear hysteretic system to an excitation with approximately white‐power spectral density and approximately Gaussian probability distribution are determined, using electronic‐analog techniques. Results are presented for the mean‐squared amplitude, the power spectral density, and the probability distribution of the response. The applicability of the Krylov‐Bogoliubov method of equivalent linearization to this problem is investigated by comparing predicted and experimentally measured values of the mean‐squared level of response.

Detection Performance of a Mean-Level Threshold

Abstract: The problem of detecting signals in nonstationary clutter is met by presenting a mean-level or adaptive threshold which adjusts to the changing background level. Such a threshold performs better than a fixed threshold that must be set for the highest amplitude clutter. However, the mean-level threshold does not perform as well for stationary noise as a fixed threshold set at the proper value. One measure of effectiveness of an adaptive threshold is its performance in stationary noise (compared to the optimum fixed threshold) for a specified speed of response. For the mean-level threshold, a simple mathematical solution is found for the detection probability when the noise is stationary and the signal scintillates rapidly. The performance is evaluated for a wide range of mean-level-threshold time constants and for several false-alarm probabilities. The results are presented graphically. As an example, the mean-level threshold suffers 3 dB in detectability (equivalent signal-to-noise ratio) in the presence of stationary noise as compared to the optimum fixed threshold for 50-percent probability of detection, false-alarm probability of 10-8, and an adjustment time of 15 times the signal duration.

Computer Models for Reservoir Regulation

Abstract: A stochastic linear programming model is structured for defining reservoir-operating policies which satisfy either single or multiple objectives. The discussion is limited to models for regulating a single reservoir subjected to random serially correlated inflows. The probabilistic sequences of inflows are defined by Markov chains. A computer program is outlined for both structuring the model and putting it in a format suitable for solution by the IBM Mathematical Programming System code. An example, based on a simplified version of one of the Finger Lakes in New York State, is used to illustrate both how the policy and the probability distributions of any function of lake levels and discharges can be obtained from the solution of the models. These probability distributions may be used to estimate the risks of flooding or droughts that may occur given a particular policy. This information should be of assistance in planning lake management policies.

A system of models for the life cycle of a biological organism

Abstract: SUMMARY This paper is concerned with stochastic models for the representation of the development of a biological organism through recognizable distinct stages. The models are based upon the assumption that the periods spent in each stage of development, excluding the possibility of death, are independent random variables with a characteristic form of probability distribution. Within each stage the organism is liable to be taken by predators or to die from other causes, and incidents of this type are assumed to occur as events in a Poisson process. The development of the theory for various forms of distribution of the period spent in a given stage is considered, including the negative exponential and second and third order special Erlangian distributions. An example is given of the application of the proposed models to the analysis of sampling data from a study of the life cycle of the grasshopper, Corthippus parallelus. The main features of the life cycle of a biological organism exhibit a similar pattern over a wide variety of different types and species. The birth of the organism occurs at a clearly defined point in time and the organism then passes through a period of growth and development until it reaches maturity. In certain types of organism this process is characterized by transition through a number of distinct and easily recognizable states in turn. An insect, for example, passes through a succession of larval instars. In other types of organism the process of development is less well defined, although it is usually possible to divide the life cycle into discrete states by reference to the presence or absence or to the size of characteristic features of the organism. At every moment of its life the organism is liable to suffer death, either as a result of the action of predators, of accidents or for other reasons. If the organism does reach maturity, its life will eventually be terminated, either by natural or other causes. In order to carry out quantitative studies of a population of a given type of organism it is often helpful to set up a mathematical model to represent the process of birth, development and death. Since there will generally be variations from one organism to another within the same population, such a model must preferably embody a stochastic or random element, as the assessment of individual variability will form an essential part of the description of the life cycle. The main features which must be taken into account are the distributions of the times of birth, of the periods spent in each stage of development and of mortality in the various stages. The process of growth, as revealed by the size of the organism at any given stage in its development, may also be of interest. This paper is concerned with a model which was originally developed to describe the life cycle of the grasshopper, Corthippus parallelus. The model is, however, of more general application, not only to other biological organisms, but also to studies of the structure of human populations. For example, the 'population' may correspond to a large organization, 'birth' may correspond to the recruit

Selection and ordering of feature observations in a pattern recognition system

Abstract: A method is proposed to aid the designer in selecting and ordering the feature observations for the pattern recognition system, without requiring the computation of the probability of misrecognition or the complete knowledge of the probability distribution for the input patterns under consideration. The essential viewpoint is basically that of pre-weighting the feature observations according to their relative importance in describing the input patterns, regardless of the specific decision structure in a recognition system. “Relative importance≓ is defined in the sense of (1) committing less error when the representation of patterns is subject to approximation due to the finite number of feature observations, and (2) carrying more information regarding the discrimination of pattern classes. It is found that, with these defined criteria, an optimal selection and ordering procedure can be obtained by establishing a generalized Karhunen-Lo\`eve coordinate system for the stochastic processes describing the input patterns. Necessary and sufficient conditions for a set of coordinates to be the generalized K\2-L system are derived and the resulting selection and ordering procedure is given. Computer-simulated experiments in character recognition are presented to illustrate the effectiveness of this method.

On Expediency and Convergence in Variable-Structure Automata

Abstract: A stochastic automaton responds to the penalties from a random environment through a reinforcement scheme by changing its state probability distribution in such a way as to reduce the average penalty received. In this manner the automaton is said to possess a variable structure and the ability to learn. This paper discusses the efficiency of learning for an m-state automaton in terms of expediency and convergence, under two distinct types of reinforcement schemes: one based on penalty probabilities and the other on penalty strengths. The functional relationship between the successive probabilities in the reinforcement scheme may be either linear or nonlinear. The stability of the asymptotic expected values of the state probability is discussed in detail. The conditions for optimal and expedient behavior of the automaton are derived. Reduction of the probability of suboptimal performance by adopting the Beta model of the mathematical learning theory is discussed. Convergence is discussed in the light of variance analysis. The initial learning rate is used as a measure of the overall convergence rate. Learning curves can be obtained by solving nonlinear difference equations relating the successive expected values. An analytic expression concerning the convergence behavior of the linear case is derived. It is shown that by a suitable choice of the reinforcement scheme it is possible to increase the separation of asymptotic state probabilities.

On the Probability of Error for Multichannel Reception of Binary Signals

Abstract: In this paper a derivation of the probability of error which arises in 1) adaptive multichannel reception of binary signals and 2) multichannel communication with binary signaling over channels that are characterized by both a specular (nonfading or constant) component and a Rayleigh fading component is presented.

Infinite dams with inputs forming a Markov chain

Abstract: This paper considers a finite dam fed by inputs forming a Markov chain. Relations for the probability of first emptiness before overflow and with overflow are obtained and their probability generating functions are derived; expressions are obtained in the case of a three state transition probability matrix. An equation for the probability that the dam ever dries up before overflow is derived and it is shown that the ratio of these probabilities is independent of the size of the dam. A time dependent formula for the probability distribution of the dam content is also obtained.

Nonparametric Discrimination Using Tolerance Regions

Abstract: Nonparametric discrimination among distributions on Euclidean space with continuous distribution functions by tolerance regions method, emphasizing errors probability distribution control

Uniform asymptotic expansions for saddle point integrals — Application to a probability distribution occurring in noise theory

Abstract: The noncentral chi-square distribution occurs in noise interference problems. When the number of degrees of freedom becomes large, the middle portion of the distribution is given by the central limit theorem, and the tails by a classical saddle point expansion. Here recent work by N. Bleistein and F. Ursell on “uniform” asymptotic expansions is combined and extended to obtain an asymptotic series which apparently holds over the entire range of the distribution. General methods for expanding saddle point integrals in uniform asymptotic series are discussed. Recurrence relations are given for the coefficients in two typical cases, (i) when there are two saddle points and (ii) when there is only one saddle point but it lies near a pole or a branch point.

The hyperlang probability distribution- a generalized traffic headway model

Abstract: THIS RESEARCH IS CONCERNED WITH THE DEVELOPMENT OF THE HYPERLANG PROBABILITY DISTRIBUTION AS A GENERALIZED TIME HEADWAY MODEL FOR SINGLE-LANE TRAFFIC FLOWS ON TWO-LANE, TWO-WAY ROADWAYS. THE STUDY METHODOLOGY INVOLVED A PROCESS THAT IS BEST DESCRIBED AS MODEL EVOLUTION, AND INCLUDED: (1) IDENTIFICATION OF SALIENT HEADWAY PROPERTIES, (2) CONSTRUCTION OF MATHEMATICAL MICRO-COMPONENTS TO SIMULATE ESSENTIAL HEADWAY PROPERTIES, (3) INTEGRATION OF THE MICRO- COMPONENTS INTO A GENERAL MATHEMATICAL HEADWAY MODEL, (4) NUMERICAL EVALUATION OF MODEL PARAMETERS, AND (5) STATISTICAL EVALUATION OF THE MODEL. THE PROPOSED HYPERLANG HEADWAY MODEL IS A LINEAR COMBINATION OF A TRANSLATED EXPONENTIAL FUNCTION AND A TRANSLATED ERLANG FUNCTION. THE EXPONENTIAL COMPONENT OF THE DISTRIBUTION DESCRIBES THE FREE (UNCONSTRAINED) HEADWAYS IN THE TRAFFIC STREAM, AND THE ERLANG COMPONENT DESCRIBES THE CONSTRAINED HEADWAYS. IT IS A VERY FLEXIBLE MODEL THAT CAN DECAY TO A SIMPLE EXPONENTIAL FUNCTION, TO AN ERLANG FUNCTION, OR TO A HYPER-EXPONENTIAL FUNCTION AS MIGHT BE REQUIRED BY THE TRAFFIC SITUATION. IT IS LIKELY, HOWEVER, THAT A TRAFFIC STREAM WILL ALWAYS CONTAIN BOTH FREE AND CONSTRAINED VEHICLES, AND THAT, THE GENERAL FORM OF THE HYPERLANG FUNCTION WILL BE REQUIRED IN ORDER TO EFFECT AN ADEQUATE DESCRIPTION OF THE COMPOSITE HEADWAYS. THE PARAMETERS OF THE HYPERLANG FUNCTION WERE EVALUATED FOR DATA SETS OBTAINED FROM THE 1965 HIGHWAY CAPACITY MANUAL AND FROM A 1967 PURDUE UNIVERSITY RESEARCH PROJECT. THE PROPOSED HYPERLANG MODEL PROVED TO BE A SOUND DESCRIPTOR OF THE REPORTED HEADWAYS FOR VOLUMES RANGING FROM ABOUT 150 VPH TO ABOUT 1050 VPH. HOWEVER, IT SHOULD BE SUBSTANTIATED AND EVALUATED FOR A WIDE RANGE OF TRAFFIC AND ROADWAY CONDITIONS. DURING THE CONDUCT OF ANY FUTURE RESEARCH, CAREFUL ATTENTION SHOULD BE GIVEN TO PROPER FLOW RATE MONITORING AND TO PROPER DATA STRATIFICATION TO REFLECT THE VARIATIONS IN HEADWAY CHARACTERISTICS THAT ARE CAUSED BY VARIATIONS IN TRAFFIC AND ROADWAY CONDITIONS. /AUTHOR/

Closed Finite Queuing Networks with Time Lags

Abstract: Consider a closed finite queuing system in which m units move among stations 1, 2, ', NN ≧ 1. A unit that completes service at station i moves to station ji, j = 1, 2, ', N with probability ej∣i ∑j = Nj =1ej∣i=1, i = 1, 2, ', N where it either commences service immediately if there is a server available, or queues to await service. The time taken for a unit to move from station i to station j is a random variable with distribution function Gi,j⋯. At each station, the service times are independent and exponentially distributed, with the instantaneous service rate at a station being an arbitrary function of the number of units at that station. The positions of the units enroute between stations, measured in time units, are introduced as supplementary variables yielding a Markov process with states having a combination of discrete and continuous components. The integro-differential difference equations for the state variables are derived, and the steady-state solution is determined. The marginal joint distribution for the number of units at each station is found, and shown to depend, not on the distributions of the time lags, but rather on some linear combination of the means of the time lag distributions. An approximate technique is outlined for determining the absolute probability distribution of the number of units at a station, and some of the limitations of this method are discussed.

Some probability distributions for neutron transport in a half-space

Abstract: We will consider a problem which arises in the theories of neutron transport and radiative (photon) transfer. We will consider, in a particular case, the random motion of a particle in a scattering medium and determine asymptotics for the probability distribution of the random variable for the number of collisions before exit from the medium.

Seismological measurements and measurement error

Abstract: A seismological measurement, such as arrival time or, less directly origin time, is an example of a measurement variable which can be considered as the sum of a parameter—the quantity being measured—and an error variable. Optimal methods for the estimation of this parameter vary with the probability distribution of the error variable. In particular, estimation in the presence of bias or of gross errors is discussed, together with the related problem of precision versus accuracy of the estimate. Errors in estimates of arrival times, origin times and hypocentral location contribute to variation in travel-time estimates; these are analyzed separately. Each of these, with the exception of focal depth, has a distribution which can be fitted to a mixture of a normal distribution and some contamination. The degree of contamination varies; methods for truncation are suggested. The presence of possible, often undetectable, bias in locations and travel times may make confidence statements about these parameters unreliable.

Distributions Determined by Cutting a Simplex with Hyperplanes

Abstract: : A geometric derivation is given for the probability that a uniformly distributed random point in a simplex shall be on one side of a given hyperplane. The argument extends directly to several hyperplanes. (Author)

Suboptimal algorithms for the quadratic assignment problem

Abstract: The idea of combining relatively simple continuous methods with discrete procedures is used for the construction of suboptimal algorithms for quadratic assignment problems. Depending on the nature of the special problem these steps may vary in complexity. The simplest procedures require minimum storage space and result in tolerable computation times. Different choices of parameters and random variations may be used in order to obtain statistical distributions of suboptimal solutions. Computational results for sample problems indicate improvements on results of Steinberg, Gilmore, and Hillier and Connors.

Portfolio distributions and tests of security selection models

Abstract: KNOWLEDGE OF THE PROBABILITY DISTRIBUTIONS of common stock returns is useful and quite often necessary both as input information for procedures for asset selection and as a standard of comparison for evaluating the output of these models of asset selection. This holds for applied as well as theoretical models or procedures. The first half of this study discusses the relation between certain portfolio decision models and probability distribution of returns to portfolios. In these sections, use is made of empirically generated probability distributions of common-stock returns to evaluate two widely used assumptions regarding the probability basis for portfolio decision models. It will be shown that these assumptions are at least open to question on a statistical basis. The second half of the study is concerned with using the empirically derived distributions as a standard of comparison against which results from actual portfolio decision rules may be tested. We have, as examples of this testing procedure, evaluated the performance of a small group of mutual funds, and the performance of several simple computerized security selection techniques.