Showing papers on "Probability distribution published in 1978"

Probability and Statistics for Engineers and Scientists.

Abstract: 1. Introduction to Statistics and Data Analysis 2. Probability 3. Random Variables and Probability Distributions 4. Mathematical Expectations 5. Some Discrete Probability Distributions 6. Some Continuous Probability Distributions 7. Functions of Random Variables (optional) 8. Fundamental Distributions and Data Description 9. One and Two Sample Estimation Problems 10. One and Two Sided Tests of Hypotheses 11. Simple Linear Regression 12. Multiple Linear Regression 13. One Factor Experiments: General 14. Factorial Experiments (Two or More Factors) 15. 2k Factorial Experiments and Fractions 16. Nonparametric Statistics 17. Statistical Quality Control 18. Bayesian Statistics

Temporal resolution of uncertainty and dynamic choice theory

Abstract: We consider dynamic choice behavior under conditions of uncertainty, with emphasis on the timing of the resolution of uncertainty. Choice behavior in which an individual distinguishes between lotteries based on the times at which their uncertainty resolves is axiomatized and represented, thus the result is choice behavior which cannot be represented by a single cardinal utility function on the vector of payoffs. Both descriptive and normative treatments of the problem are given and are shown to be equivalent. Various specializations are provided, including an extension of "separable" utility and representation by a single cardinal utility function. CONSIDER THE FOLLOWING idealization of a dynamic choice problem with uncertainty. At each in a finite, discrete sequence of times t = 0, 1, . . ., T, an individual must choose an action d,. His choice is constrained by what we temporarily call the state at time t, xt. Then some random event takes place, determining an immediate payoff zt to the individual and the next state xt+l. The probability distribution of the pair (zt, xt+l) is determined by the action dt. The standard approach in analyzing this problem, which we will call the payoff vector approach, assumes that the individual's choice behavior is representable as follows: He has a von Neumann-Morgenstern utility function U defined on the vector of payoffs (z0, z1, . . ., ZT). Each strategy (which is a contingent plan for choosing actions given states) induces a probability distribution on the vector of payoffs. So the individual's choice of action is that specified by any optimal strategy, any strategy which maximizes the expectation of utility among all strategies (assuming sufficient conditions so that an optimal strategy exists). This paper presents an axiomatic treatment of the dynamic choice problem which is more general than the payoff vector approach, but which still permits tractable analysis. The fundamental difference between our treatment and the payoff vector approach lies in our treatment of the temporal resolution of uncertainty: In our models, uncertainty is "dated" by the time of its resolution, and the individual regards uncertainties resolving at different times as being different. For example, consider a situation in which a fair coin is to be flipped. If it comes up heads, the payoff vector will be (zo, z1) = (5, 10); if it is tails, the vector will be (5, 0). Because z0 = 5 in either case, the coin flip can take place at either time 0 or time 1. It will not matter when the flip occurs to someone who has cardinal utility on the vector of payoffs. But an individual can obey our axioms and prefer either one to the other. One justification for our approach is the well known "timeless-temporal" or "joint time-risk" feature of some models (usually models which are not "complete"). For example, preferences on income streams which are induced from primitive preferences on consumption streams in general depend upon when the

Estimation of the size of a closed population when capture probabilities vary among animals

Abstract: SUMMARY A model which allows capture probabilities to vary by individuals is introduced for multiple recapture studies on closed populations. The set of individual capture probabilities is modelled as a random sample from an arbitrary probability distribution over the unit interval. We show that the capture frequencies are a sufficient statistic. A nonparametric estimator of population size is developed based on the generalized jackknife; this estimator is found to be a linear combination of the capture frequencies. Finally, tests of underlying assumptions are presented.

On the existence of probability measures with given marginals

Abstract: © Annales de l’institut Fourier, 1978, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Gaussian Random Processes

Abstract: I Preliminaries.- I.1 Gaussian Probability Distribution in a Euclidean Space.- I.2 Gaussian Random Functions with Prescribed Probability Measure.- I.3 Lemmas on the Convergence of Gaussian Variables.- I.4 Gaussian Variables in a Hilbert Space.- I.5 Conditional Probability Distributions and Conditional Expectations.- I.6 Gaussian Stationary Processes and the Spectral Representation.- II The Structures of the Spaces H(T) and LT(F).- II. 1 Preliminaries.- II.2 The Spaces L+(F) and L-(F).- II.3 The Construction of Spaces LT(F) When T Is a Finite Interval.- II.4 The Projection of L+(F) on L-(F).- II.5 The Structure of the ?-algebra of Events U(T).- III Equivalent Gaussian Distributions and their Densities.- III.1 Preliminaries.- III.2 Some Conditions for Gaussian Measures to be Equivalent.- III.3 General Conditions for Equivalence and Formulas for Density of Equivalent Distributions.- III.4 Further Investigation of Equivalence Conditions.- IV Conditions for Regularity of Stationary Random Processes.- IV.1 Preliminaries.- IV.2 Regularity Conditions and Operators Bt.- IV.3 Conditions for Information Regularity.- IV.4 Conditions for Absolute Regularity and Processes with Discrete Time.- IV.5 Conditions for Absolute Regularity and Processes with Continuous Time.- V Complete Regularity and Processes with Discrete Time.- V.l Definitions and Preliminary Constructions with Examples.- V.2 The First Method of Study: Helson-Sarason's Theorem.- V.3 The Second Method of Study: Local Conditions.- V.4 Local Conditions (continued).- V.5 Corollaries to the Basic Theorems with Examples.- V.6 Intensive Mixing.- VI Complete Regularity and Processes with Continuous Time.- VI.1 Introduction.- VI.2 The Investigation of a Particular Function ?(T ).- VI.3 The Proof of the Basic Theorem on Necessity.- VI.4 The Behavior of the Spectral Density on the Entire Line.- VI.5 Sufficiency.- VI.6 A Special Class of Stationary Processes.- VII Filtering and Estimation of the Mean.- VII.1 Unbiased Estimates.- VII.2 Estimation of the Mean Value and the Method of Least Squares.- VII.3 Consistent Pseudo-Best Estimates.- VII.4 Estimation of Regression Coefficients.- References.

The Strong Limits of Random Matrix Spectra for Sample Matrices of Independent Elements

Abstract: This paper proves almost-sure convergence of the empirical measure of the normalized singular values of increasing rectangular submatrices of an infinite random matrix of independent elements. The limit is the limit as both dimensions grow large in some ratio. The matrix elements are required to have uniformly bounded central $2 + \delta$th moments, and the same means and variances within a row. The first section (relaxing the restriction on variances) proves any limit-in-distribution to be a constant measure rather than a random measure, establishes the existence of subsequences convergent in probability, and gives a criterion for almost-sure convergence. The second section proves the almost-sure limit to exist whenever the distribution of the row variances converges. It identifies the limit as a nonrandom probability measure which may be evaluated as a function of the limiting distribution of row variances and the dimension ratio. These asymptotic formulae underlie recently developed methods of probability plotting for principal components and have applications to multiple discriminant ratios and other linear multivariate statistics.

Probability and Statistical Inference

Abstract: 1. Empirical and Probability Distributions. Basic Concepts. The Mean, Variance, and Standard Deviation. Continuous-Type Data. Exploratory Data Analysis. Graphical Comparisons of Data Sets. Time Sequences. Probability Density and Mass Functions. 2. Probability. Properties of Probability. Methods of Enumeration. Conditional Probability. Independent Events. Bayes' Theorem. 3. Discrete Distributions. Random Variables of the Discrete Type. Mathematical Expectation. Bernoulli Trials and the Binomial Distribution. The Moment-Generating Function. The Poisson Distribution. 4. Continuous Distributions. Random Variables of the Continuous Type. The Uniform and Exponential Distributions. The Gamma and Chi-Square Distributions. The Normal Distribution. Distributions of Functions of a Random Variable. Mixed Distributions and Censoring. 5. Multivariable Distributions. Distributions of Two Random Variables. The Correlation Coefficient. Conditional Distributions. The Bivariate Normal Distribution. Transformations of Random Variables. Order Statistics. 6. Sampling Distribution Theory. Independent Random Variables. Distributions of Sums of Independent Random Variables. Random Functions Associated with Normal Distributions. The Central Limit Theorem. Approximations for Discrete Distributions. The t and F Distributions. Limiting Moment-Generating Functions. Chebyshev's Inequality and Convergence in Probability. Importance of Understanding Variability. 7. Estimation. Point Estimation. Confidence Intervals for Means. Confidence Intervals for Difference of Two Means. Confidence Intervals for Variances. Confidence Intervals for Proportions. Sample Size. Distribution-Free Confidence Intervals for Percentiles. A Simple Regression Problem. More Regression. 8. Tests of Statistical Hypotheses. Tests about Proportions. Tests about One Mean and One Variance. Tests of the Equality of Two Normal Distributions. Chi-Square Goodness of Fit Test. Contingency Tables. Tests of the Equality of Several Means. Two-Factor Analysis of Variance. Tests Concerning Regression and Correlation. The Wilcoxon Tests. Kolmogorov-Smirnov Goodness of Fit Test. Resampling Methods. Run Test and Test for Randomness. 9. Theory of Statistical Inference. Sufficient Statistics. Power of a Statistical Test. Best Critical Regions. Likelihood Ratio Tests. Bayesian Estimation. Asymptotic Distributions of Maximum Likelihood Estimators. 10. Quality Improvement through Statistical Methods. Statistical Quality Control. General Factorial and 2k Factorial Designs. More on Design of Experiments. Epilogue.Appendix A. Review of Selected Mathematical Techniques. Algebra of Sets. Mathematical Tools for the Hypergeometric Distribution. Limits. Infinite Series. Integration. Multivariate Calculus. Appendix B. References.Appendix C. Tables.Appendix D. Answers to Odd-Numbered Exercises.Index.

An Analysis of Competing Software Reliability Models

Abstract: This paper examines the most widely used reliability models. The models discussed fall into two categories, the data domain and the time domain. Besides tracing the historical development of the various models their advantages and disadvantages are analyzed. This includes models based on discrete as weil as continuous probability distributions. How well a given model performs its purpose in a specific economic environment will determine the usefulness of the model. Each of the models is examined with actual data as to the applicability of the error fmding process.

The problem of the Nile: Conditional solution to a changepoint problem

Abstract: SUMMARY Inference is considered for the point in a sequence of random variables at which the probability distribution changes. An approximation to the conditional distribution of the maximum likelihood estimator of the changepoint given the ancillary values of observations adjacent to the estimated changepoint is derived and shown to be numerically equal to a Bayesian posterior distribution for the changepoint. A hydrological example is given to show that inferences based on the conditional distribution of the maximum likelihood estimator can differ sharply from inferences based on the marginal distribution. the process governing their distribution changes abruptly, and consider the problem of inference about the unknown changepoint. Published research on this and related problems has provided changepoint estimators for a class of increasingly sophisticated models; for a recent example and references, see Ferriera (1975). The present paper turns back the clock to reconsider the simplest possible changepoint problem, one involving independent random variables whose distributions are completely specified apart from the unknown changepoint. A Bayesian solution to this problem is implicit in the work of Chernoff & Zacks (1964); a frequentist solution is given by Hinkley (1970). We consider here a third solution, based on a conditional frequentist approach. In a sense to be made precise, this third solution serves as a bridge linking the previous two. The conditional solution evolves from Hinkley's frequentist approach, which bases inferences on the asymptotic sampling distribution of the maximum likelihood estimator of the changepoint. The need for conditioning arises because the maximum likelihood estimator is not a sufficient statistic, and thus inferences based on its sampling distribution can be made more informative by conditioning on the values of appropriate ancillary statistics. It turns out that for the simple changepoint problem the resulting conditional inferences are nominally equivalent to certain Bayesian inferences in the sense that numerical differences can be made arbitrarily small. Nominal equivalence of the two solutions follows from an approximate version of a result obtained by Fisher (1934) in his conditional approach to estimating a translation parameter 9: if A is ancillary in that its distribution does not depend on 0, and if the density f(x I 0) of the data x can be factorized in the form

Statistical Treatment of Experimental Data

Abstract: 1. Introduction. 2. Probability. 3. Random Variables and Sampling Distributions. 4. Some Important Probability Distributions. 5. Estimation. 6. Confidence Intervals. 7. Hypothesis Testing. 8. Tests on Means. 9. Tests on Variances. 10. Goodness of Fit Tests. 11. Correlation. 12. The Straight Line Through the Original or Through Some Other Fixed Point. 13. The Polynomial Through the Origin or Through Some Other Fixed Point. 14. The General Straight Line. 15. The General Polynomial. 16. A Brief Look at Multiple Regression. Appendices: 1. Drawing a Random Sample Using a Table of Random Numbers. 2. Orthogonal Polynomials in x. References. Index.

On the entropy of continuous probability distributions (Corresp.)

Abstract: A table is given of differential entropies for various continuous probability distributions. The formulas, some of which are new, are of use in the calculation of rate-distortion functions and in some statistical applications.

Urn Models and Their Application—An Approach to Modern Discrete Probability Theory

Abstract: (1978). Urn Models and Their Application—An Approach to Modern Discrete Probability Theory. Technometrics: Vol. 20, No. 4, pp. 501-501.

Statistics of normal mode amplitudes in a random ocean. I. Theory

Abstract: A statistical theory of acoustic propagation in a model random ocean, valid in the limit of low acoustic frequency, is presented. A random internal‐wave model gives sound‐speed fluctuations δc (r,z,t) about a deterministic profile ? (z). Using normal modes of ? (z) as a basis, the theory gives quantitative estimates of statistical moments of the mode amplitudes ψn(r,t), which are randomly coupled via δc. Invoking a quasistatic approximation, the theory treats time as a parameter. From any initial (r=0) distribution of modal powers ‖ψn‖2, the evolution of their averages to an equilibrium is predicted by ’’coupled power’’ equations. The theory makes similar predictions for average fluctuations of the modal powers about their means. In the equilibrium limit, the theory gives the full probability distribution of the ψn.

The Statistics of Curie-Weiss Models

Abstract: LetSn denote the random total magnetization of ann-site Curie-Weiss model, a collection ofn (spin) random variables with an equal interaction of strength 1/n between each pair of spins. The asymptotic behavior for largen of the probability distribution ofSn is analyzed and related to the well-known (mean-field) thermodynamic properties of these models. One particular result is that at a type-k critical point (Sn-nm)/n1−1/2k has a limiting distribution with density proportional to exp[-λs2k/(2k)!], wherem is the mean magnetization per site and A is a positive critical parameter with a universal upper bound. Another result describes the asymptotic behavior relevant to metastability.

Distribution of membrane thickness determined by lineal analysis.

Abstract: SUMMARY The expected statistical distributions of intercept length are derived in terms of geometrical probability density functions pertaining to plates with known thickness penetrated by lines with random orientation. These expressions provide arithmetic and graphical solutions for obtaining distributions of membrane thickness and reciprocal membrane thickness from empirical distributions of intercept lengths. Furthermore, general relationships between probability density functions of distributions of intercept length and membrane thickness are derived as well as those between their moments. Examples of the application of the method to biological samples are given, and estimated distributions of glomerular basement membrane thickness are compared to those obtained by an independent, direct method. Various sources of bias, which in practice may occur due to departures from the sample model, are discussed and the influence of some of them is estimated. The knowledge of the probability density function of reciprocal intercepts makes it possible to perform a correction of the distributions of measured intercept length, which to some extent eliminates bias.

Approximate Evaluation of Detection Probabilities in Radar and Optical Communications

Abstract: Cumulative probability distributions such as occur in radar detection problems are approximated by a new version of the saddlepoint method of evaluating the inverse Laplace transform of the moment-generating function. When the number of radar pulses integrated is large, the approximation of lowest order yields good accuracy in the tails of the distributions, yet requires much less computation than standard recursive methods. Greater accuracy can be achieved upon summing the residual series by converting it to a continued fraction. The method is applied to evaluating the error-function integral and the Mth-order Q function, and to approximating the inverse of the chi-squared distribution. Cumulative distributions of discrete random variables, needed for determining error probabilities in optical communication receivers that involve counting photoelectrons, can be approximated by a simple modification of the method, which is here applied to the Laguerre distribution.

Growth of fluctuations in quenched time-dependent Ginzburg-Landau model systems

Abstract: A dynamical theory is presented for the enhanced fluctuations that occur in a time-dependent Ginzburg-Landau model system with the order parameter not conserved which is quenched from a thermodynamically stable to an unstable state. In a certain weak-coupling, long-time, and long-distance limit, diffusion and saturation effects can be treated separately. As a result explicit expressions are found for the probability distribution functional, the two-point reduced distribution function, and the pair correlation function of the fluctuations, which evolve from an arbitrary initial probability distribution functional. The behavior of the latter two functions is also displayed graphically. A central role is played by the time-independent nonlinear transformation of the order parameter which takes care of the saturation effects. The nature of such a transformation is discussed in a general context. If the problem is viewed as a nonequilibrium critical phenomenon, the theory corresponds to the Landau mean-field theory. An expansion in $\ensuremath{\epsilon}=4\ensuremath{-}d$ is suggested to improve our treatment, where $d$ is the dimensionality of space.

Probabilistic Methods For Power System Dynamic Stability Studies

Abstract: In this paper, several sources of uncertainty in electric power systems are incorporated into the dynamic stability analysis of the system. Operating point stability is considered, with the system model written in state variable notation. The sensitivites of the eigenvalues of the associate matrix are used to calculate the statistics of eigenvalue locations. When the uncertainties considered are approximated by the multivariate normal distribution, the probability of dynamic stability is computed using the generalized tetrachoric series. The principle advantages of this method over multiple runs of a deterministic stability study are rapid calculation times and the availability of consistently calculated probability of operating point stability figures.

Characterizations of Probability Distributions: A Unified Approach With an Emphasis on Exponential and Related Models

Abstract: Preliminaries and basic results.- Characterizations based on truncated distributions.- Characterizations by properties of order statistics.- Characterizations of the poisson process.- Characterizations of multivariate exponential distributions.- Miscellaneous results.

Decentralized multicriteria optimization of linear stochastic systems

Abstract: By adopting a decision-theoretic approach and under the noncooperative equilibrium solution concept of game theory, decentralized multicriteria optimization of stochastic linear systems with quasiclassical information patterns is discussed. First, the static M -person quadratic decision problem is considered, and sufficiency conditions are derived for existence of a unique equilibrium solution when the primitive random variables have a priori known but arbitrary probability distributions with finite second-order moments. The optimal strategies are given in the form of the limit of a convergent sequence which is shown to admit a closed-form linear solution for the special case of Gaussian distributions. Then, this result is generalized to dynamic LQG problems, and a general theorem is proven, which states that under the one-step-delay observation sharing pattern this class of systems admit unique affine equilibrium solutions. This result, however, no longer holds true under the one-step-delay sharing pattern, and additional criteria have to be introduced in this case. These results are then interpreted within the context of LQG team problems, so as to generalize and unify some of the results found in the literature on team problems.

Probabilistic formulation of the emergency service location problem.

Abstract: The problem of locating emergency service facilities is studied under the assumption that the locations of incidents (accidents, fires, or customers) are random variables. The probability distribution for rectilinear travel time between a new facility location and the random location of the incident Pi is developed for the case of Pi being uniformly distributed over a rectangular region. The location problem is considered in a discrete space. A deterministic formulation is obtained and recognized to be a set cover problem. Probabilistic variation of the central facility location problem is also presented.

Numerical methods for stochastic processes

Abstract: Two numerical methods aimed at discrete state continuous time stochastic processes are discussed. The first (inversion) method is applicable to processes determined by the equation for their probability transform. For instance, many Markov processes are determined by a partial differential equation. The intractability of the presently available analytic methods is overcome by requiring only a numerical solution to the transform equation. Applying the inverse Fourier transform to this solution then yields an approximation to the distribution of a process at an epoch. Useful error bounds are presented. The second (recursion) method, which can be used in conjunction with the first, is of much greater generality, but less efficient. By recursively applying the Chapman- Kolmogorov equation to some initial distribution, the probability distribution for any sequence of epochs can be computed, hence giving a complete description of a process through time. This method can be applied to absorbing processes, and it is shown that the order of the approximation is O(h).

A Problem in Multivariate Statistics: Algorithm, Data Structure and Applications.

Abstract: : Problems and applications are investigated which are associated with computing the empirical cumulative distribution function of N points in k-dimensional space and a multidimensional divide-and-conquer technique is employed that gives rise to a compact data structure for geometric and statistical search problems. A large number of important statistical quantities are computed much faster than was previously possible.

On a Probability Bound of Marshall and Olkin

Abstract: A counterexample is given to a proposition of Marshall and Olkin (1974, Ann. Statist.) describing a probability bound. A theorem which gives the conditions under which mixtures of associated random variables remain associated, is stated. This provides a method to obtain the required bound.

Lecture notes on medical statistics

Abstract: Contents: Introduction Descriptive statistics Probability Probability distributions Sampling and sampling distributions Estimation Tests of hypotheses Comparison of two means and two variances Analysis of variance The chi-square test Linear regression and correlation Distribution-free methods Clinical trials Vital statistics Observational studies.