Showing papers on "Probability mass function published in 1985"

A data distortion by probability distribution

Abstract: This paper introduces data distortion by probability distribution, a probability distortion that involves three steps. The first step is to identify the underlying density function of the original series and to estimate the parameters of this density function. The second step is to generate a series of data from the estimated density function. And the final step is to map and replace the generated series for the original one. Because it is replaced by the distorted data set, probability distortion guards the privacy of an individual belonging to the original data set. At the same time, the probability distorted series provides asymptotically the same statistical properties as those of the original series, since both are under the same distribution. Unlike conventional point distortion, probability distortion is difficult to compromise by repeated queries, and provides a maximum exposure for statistical analysis.

Poisson boundaries of discrete groups of matrices

Abstract: If μ is a probability measure on a countable group there is defined a notion of the Poisson boundary for μ which enables one to represent all bounded μ-harmonic functions on the group. It is shown that for discrete groups of matrices this boundary can be identified with the boundary of the corresponding Lie group.

Global asymptotic stability of the size distribution in probabilistic models of the cell cycle.

Abstract: Probabilistic models of the cell cycle maintain that cell generation time is a random variable given by some distribution function, and that the probability of cell division per unit time is a function only of cell age (and not, for instance, of cell size). Given the probability density, f(t), for time spent in the random compartment of the cell cycle, we derive a recursion relation for φ n(x), the probability density for cell size at birth in a sample of cells in generation n. For the case of exponential growth of cells, the recursion relation has no steady-state solution. For the case of linear cell growth, we show that there exists a unique, globally asymptotically stable, steady-state birth size distribution, φ *(x). For the special case of the transition probability model, we display φ *(x) explicitly.

Discrete problems in probability theory

Abstract: A short survey is given of some directions in probability theory that have developed most intensively in recent years. Separable statistics and criteria for the verification of statistical hypotheses based on them, various schemes for distributing particles among cells, and problems connected with estimating the unknown size of a finite collection are considered.

Robust Bayesian Bounds for Monetary‐Unit Sampling in Auditing

Abstract: Mixture distributions combining a probability mass at zero and a continuous density function for positive outcomes are frequently found in auditing. The Cox and Snell bound for evaluating the results of monetary unit sampling is a Bayesian bound utilizing prior information designed for such mixture distributions. In this paper it is shown that conservative prior parameter values for the Cox and Snell bound can be found such that this bound possesses classical confidence properties in repeated sampling from a wide variety of possible realized populations.

On the computation of bi-normal radial error

Abstract: The Bi-normal density distribution function on a surface is represented by a position vector and covariance matrix. Its physical dimensions are described by the error ellipse. A generalized scalar is the radial or circular error which denotes the probability within a radius of the position. To compute the radial error probability (or probability circle) precisely, a non-trivial numerical integration is necessary. Simpler but less accurate conventions in common use are the Drms and CEP. The error ellipse semi-major axis is also sometimes applied to radial error. These three measures of radial error are subject to variations in probability as a function of the eccentricity of the distribution. The probability of a circle can be obtained simply and more accurately by the use of a third order polynomial.

Exact joint probability distribution for centrosymmetric structure factors. Derivation and application to the Φ1relationship in the space groupPgif">\overline{1}

Abstract: Recent applications of exact random-walk techniques to crystallographic structure-factor statistics have now been extended to multivariate joint probability density functions of several structure factors. The technique of deriving such multivariate exact density functions is introduced, and is applied to the study of the simplest sign relationship: Σ1, in the space group P\bar 1. An exact expression is obtained for the probability that the sign of the normalized structure factor E(2h 2k 2l) is positive, given the magnitudes of E(2h 2k 2l) and E(h k l), and an extensive numerical examination of this new expression - as compared with the conventional asymptotic formula for this probability - is presented. It is shown that the asymptotic formula usually underestimates the probability that E(2h 2k 2l) is positive, the discrepancies between the exact and asymptotic results being rather serious when the atomic composition of the asymmetric unit is heterogeneous (even moderately so), and when the number of atoms is small; a paucity of atoms leads to significant discrepancies even in the equal-atom case. On the other hand, for large asymmetric units of low heterogeneity and for high E values, the exact and asymptotic expressions agree very well in their predictions. The qualitative behaviour of the new exact expression is consistent with the known features of the Σ1 relationship and its statistical interpretations.

A Probability Bound Estimation Method in Markov Reliability Analysis

Abstract: In many practical systems, the uncertainty of component failure/repair rates results in uncertainty of system failure probability. Concerning a repairable system, uncertainty is evaluated as a probability bound in the Markov process. In practical analysis, the Laplace transform has the advantage of relatively less computing time than that of a numerical method, eg, Runge Kutta. This paper proposes an algorithm for evaluating this uncertainty using the Laplace transform method. This algorithm assumes the Johnson SB distribution for system-failure probability. Then, the mean and the variance of system-failure probability are obtained using Newton's method and an integral form for calculating parametric differentiation. Finally, the probability bounds are obtained by applying the conventional moment-matching method. A tutorial example is presented at the end of this paper.

The probability generating function of empty cell variable in a randomized occupancy problem

Abstract: Let Mo denote the number of empty cells when n distinguishable balls are distributed independently and at random in ra cells such that each ball stays with probability p in its cell, and falls through with probability 1-p. We find the probability generating function of Mo by solving a partial differential equation satisfied by a suitable generating function. The corresponding function for the classical case p = 1 is well-known, but obtained by different methods.

Second‐order statistics and discrete‐time detection modeling for partially saturated processes

Abstract: The basic problem in underwater detection is formulated under the general assumptions of partially saturated propagation of narrow‐band and acoustic signals. Expressions for the joint probability density function (PDF) of ρ, the short‐time average root‐mean‐square pressure at the receiver are obtained. This joint PDF is a general result reducing to the PDF’s for the fully saturated and the unsaturated cases for limiting values of the appropriate variables. Subsequently, defining detection as occurring whenever ρ exceeds a specified threshold level ρ0 and, using the above results, the upcrossing and downcrossing statistics of the envelope process are studied. Closed form expressions for the probability mass functions (PMF’s) of the interarrival time (time between two successive detections) and holding time (time between an upcrossing and the first subsequent downcrossing) are obtained. Results using our partially saturated detection model reduce, in limiting cases, to results already obtained in the literature for fully saturated and unsaturated propagation.

An Alternative Method for Analyzing the Double Threshold Detector

Abstract: An alternative method for analyzing the performance of a double threshold or M-out-of-N detector is discussed. Detection performance for the suggested method is based on the probability that a return crosses the threshold for the Mth time (a detection is declared) on the kth return or look. It is shown that this formulation has many advantages, as compared with the conventional method of analysis which employs the binomial probability distribution, since the upper limit N is not contained in the resulting probability expressions. It is shown that the probability of detection obtained by the alternate method is the same as that obtained if the detection method were analyzed as a Markov chain with M+1 states. Use of the method results in simple expressions for the mean and variance of the number of looks before detection, provides an alternative way of estimating the probability of a threshold crossing, and leads to computationally simple bounds for the probability of false alarm.

Probability and linear system theory

Abstract: This book is concerned with the analysis of discrete-time linear systems subject to random disturbances. This introductory chapter is designed to present the main results in the two areas of probability and linear systems theory as required for the main developments of the book, beginning in Chapter 2.

Measures of location in the plane

Abstract: There are many possible candidates for measures of location of asymmetric probability distributions. This difficulty is compounded for multivariate distributions. It is the purpose of this paper to characterize the set of all possible measures of location for a given bivariate probability distribution. A closed, convex region in the plane will be constructed, any point of which is a reasonable measure of location. Reasonable here refers to the invariance of the region under certain transformations and order relations. The size of this region can be used to characterize the degree of asymmetry that a distribution possesses.

Some discrete probability distributions

Abstract: There are some theoretical probability distributions that occur sufficiently frequently in statistical work to warrant individual consideration in our text. In this chapter we discuss four discrete probability distributions: uniform, binomial, geometric and Poisson.

Carathéodory convex integrand operators and probability theory.

Abstract: In two recent papers, the author studied extensions of several concepts of nonsmooth analysis to vector valued operators. The purpose of the present work is to further continue this effort and to study, from a probabilistic viewpoint, several properties of convex operators. In particular, we will examine how various basic concepts of vectorial nonsmooth analysis associated with an integrand/(ω, x) are related to those of the integral operator F(x) = /Ω/(ω, x) dμ(ω) where the vector valued integral is defined in the sense of Bochner. Also we introduce a conditional expectation for such integrands, study several of its properties, see how it is affected by various operations of nonsmooth analysis, and derive a vector valued martingale convergence theorem.

Elements of Probability Theory

Abstract: As the Kalman-Bucy filtering theory is a probabilistic concept, an understanding of some basic concepts in probability theory is necessary in the study of this subject. We begin this discourse by reviewing some of the basic elements in probability theory. Details and proofs can be found in [1.1–-4], for example.