Showing papers on "Poisson distribution published in 1977"

The Poisson Representation. I. A New Technique for Chemical Master Equations

Abstract: We introduce a new technique for handling chemical master equations, based on an expansion of the probability distribution in Poisson distributions. This enables chemical master equations to be transformed into Fokker-Planck and stochastic differential equations and yields very simple descriptions of chemical equilibrium states. Certain nonequilibrium systems are investigated and the results are compared with those obtained previously. The Gaussian approximation is investigated and is found to be valid almost always, except near critical points. The stochastic differential equations derived have a few novel features, such as the possibility of pure imaginary noise terms and the possibility of higher order noise, which do not seem to have been previously studied by physicists. These features are allowable because the transform of the probability distribution is a quasiprobability, which may be negative or even complex.

Level Crossings in Point Processes Applied to Queues: Single-Server Case

Abstract: This paper introduces a new methodology for obtaining the stationary waiting time distribution in single-server queues with Poisson arrivals. The basis of the method is the observation that the stationary density of the virtual waiting time can be interpreted as the long-run average rate of downcrossings of a level in a stochastic point process. Equating the total long-run average rates of downcrossings and upcrossings of a level then yields an integral equation for the waiting time density function, which is usually both a linear Volterra and a renewal-type integral equation. A technique for deriving and solving such equations is illustrated by means of detailed examples.

Finding the edge of a Poisson forest

Abstract: The points of a homogeneous Poisson process within a compact convex set are observed. We consider how to reconstruct this domain from the observations.

Modification of Poisson statistics: modeling defects induced by diffusion

Abstract: This paper examines a model of LSI device failure and the departure from Poisson statistics that it necessitates. By visually mapping anodically decorated transistors, the authors found that in highly defective sites, emitter-collector shorts-pipes-tend to collect in clusters of totally defective areas. Less defective sites have a nearly random distribution of defects, though some limited clustering may still exist. In general, a slightly curved relationship is obtained when the logarithm of actual yield is plotted versus area. However, for a small enough area, such as a single chip, one can make a linear approximation and use it to estimate the fraction of the area that is totally defective, and the defect density. The paper describes an analytical method of modeling device failures, and of projecting yields for areas larger than the data base from which the parameters of the yield equation were estimated.

Sampling errors in the measurement of rain and hail parameters

Abstract: Attention is given to a general derivation of the fractional standard deviation (FSD) of any integrated property X such that X(D) = cD to the n. This work extends that of Joss and Waldvogel (1969). The equation is applicable to measuring integrated properties of cloud, rain or hail populations (such as water content, precipitation rate, kinetic energy, or radar reflectivity) which are subject to statistical sampling errors due to the Poisson distributed fluctuations of particles sampled in each particle size interval and the weighted sum of the associated variances in proportion to their contribution to the integral parameter to be measured. Universal curves are presented which are applicable to the exponential size distribution permitting FSD estimation of any parameters from n = 0 to n = 6. The equations and curves also permit corrections for finite upper limits in the size spectrum and a realistic fall speed law.

Distance distributions associated with poisson processes of geometric figures

Abstract: Consider a process of identically-shaped (but not necessarily equal-sized) figures (e.g. points, clusters of points, lines, spheres) embedded at random in n-dimensional space. A simple technique is derived for finding the distribution of the distance from a fixed point, chosen independently of the process of figures, to the k th nearest figure. The technique also shows that the distribution is independent of the distribution of the orientations of the figures. It is noted that the distribution obtained above (for equal-sized figures) is identical to the distribution of the distance from a fixed figure to the k th nearest of a random process of points.

Prediction and estimation for the compound Poisson distribution.

Abstract: An empirical Bayes method is proposed for predicting the future performance of certain subgroups of a compound Poisson population.

Poisson limits for a clustering model of strauss

Abstract: In this article we present a limiting result for the random variable Yn (r) which arises in a clustering model of Strauss (1975) The result is that under some sparseness-of-points conditions the process {Yn (r): 0 ≦ r ≦ r ∞} converges weakly to a non-homogeneous Poisson process {Y(r): 0 ≦ r ≦ r ∞} when n → ∞ Simulation results are given to indicate the accuracy of the approximation when n is moderate and applications of the limiting result to tests for clustering are discussed

A general law for direct sunlight penetration

Abstract: Under the assumption that foliage area distributions may be adequately approximated by probability distributions, a general law for the prediction of direct sunlight penetration of crop canopies is derived. It is shown that the Poisson (Beer's) law is a special case of this more general law. Area and transect problems are treated, and stochastic variation in leaf sizes is considered. Numerical studies are presented which indicate that the Poisson law is not adequate in all cases for the prediction of sunlight penetration and that stochastic variation in leaf sizes is of less consequence than average leaf size and/or leaf-area index.

Weak ₁ characterizations of Poisson integrals, Green potentials and ^{} spaces

Abstract: Our main result can be described as follows A subharmonic function u in a suitable domain 2 in Rn is the difference of a Poisson integral and a Green potential if and only if u divided by the distance to a a is in weak L1 in 2 Similar conditions are given for a harmonic function to be the Poisson integral of an L function on a a Iterated Poisson integrals in a polydisc are also considered As corollaries, we get weak L, characterizations of HP spaces of different kinds

A note on the upper bound for the distance in total variation between the binomal and the Poisson distribution

Abstract: The total variation distance between the binomial B(n, p) distribution and the Poisson P(np) distribution is smaller than 2 1/2p(1-p)-1/2 according to VERVAAT[4], [5]. We shall sharpen this inequality by using a result due to KEMPERMAN[1], CSISZAR[2] and KULLBACK[3].

Minimum variance unbiased estimation in a modified power series distribution and some of its applications

Abstract: In this paper we study the minimum variance unbiased estimation in the modified power series distribution introduced by the author (1974a). Necessary and sufficient conditions for the existence of minimum variance unbiased estimate (MVUE) of the parameter based on sufficient statistics are obtained. These results are, then, applied to obtain MVUE of θr (r ≥ 1) for the generalized negative binomial and the decapitated generalized negative binomial distributions (Jain and Consul, 1971). Similar estimates are obtained for the generalized Poisson (Consul and Jain, 1973a) and the generalized logarithmic series distributions (Jain and Gupta, 1973). Several of the well-known results follow trivially from the results obtained here.

Bayes Design of a Reservoir Under Random Sediment Yield

Abstract: The design of a reservoir subject to long-range sediment accumulation stemming from the sum of a random number of random sedimentation events is investigated. The event-based simulation method, which is applied to a case study in southern Arizona, involves generating synthetic sequences of Poisson inputs into the modified universal soil loss equation. The stochastic inputs result from a fitted bivariate distribution of runoff-producing precipitation events (representing the amount and duration of such precipitation) and an independent fitted exponential distribution of interarrival time between events. The simulated sequences of sediment yield events thus obtained are used to calculate accumulated sediment yield and cost of a given design for each sequence. The optimum design and corresponding Bayes risk are evaluated in four cases: (1) under natural uncertainty, (2) under natural uncertainty and uncertainty in the bivariate rainfall distribution parameters, (3) under natural uncertainty and uncertainty in the Poisson counting distribution parameter, and (4) under all three types of uncertainty. The effect of rainfall record length is ascertained by further computer experiments, but only a partial Bayesian analysis is provided because of the complexity created by a three-dimensional parameter uncertainty. The optimum reservoir capacity and corresponding Bayes risk are shown to increase substantially (up to 20 and 90%, respectively) as more uncertainties are incorporated into the model.

Failure Prediction for an On-Line Maintenance System in a Poisson Shock Environment.

Abstract: : An analog system subject to the Poisson Shock is modeled using past performance data. Failure Dynamics of the system is estimated by curve fitting techniques. Algorithms for fault prediction in an on-line maintenance process are described. Several sequential refinement schemes are introduced to improve fault prediction. Some formulas and properties of system's statistics have been developed. A decision rule is introduced which is based on the criteria of simultaneously maximizing lifetime and minimizing the cost of on-line failures. Poisson Shock generator is implemented by computer for simulation of the on-line maintenance process. The computer simulations of a perfect, no measurement errors and identical drifting parameters, system are presented. The simulations of an imperfect system are studied by adding a noise to the system performance data. (Author)

Testing the hypothesis that a point is Poisson

Abstract: The testing of the hypothesis that a point process is Poisson against a one-dimensional alternative is considered. The locally optimal test statistic is expressed as an infinite series of uncorrelated terms. These terms are shown to be asymptotically equivalent to terms based on the various orders of cumulant spectra. The efficiency of tests based on partial sums of these terms is found.

Multiplicative Poisson Models with Unequal Cell Rates

Abstract: A two way frequency table with independent Poisson distributed cell numbers is considered. The expected number in each cell is a product of a row effect, a column effect and a known constant. Methods are developed for estima- tion of the parameters by maximum likelihood. In addition asymptotic X2-tests are considered for checking the model and testing equality of column effects and/or row effects. The proposed method is applied to Danish lung cancer data.

Simulating nonstationary Poisson processes: a comparison of alternatives including the correct approach

Abstract: This paper describes three inexact methods that are commonly used to generate arrivals for a nonstation ary Poisson process and compares them to an exact method. Different patterns are assumed for the arrival rate as a function of time. The exact method is shown to produce accurate results, whereas all three inexact methods produce results that are sig nificantly different in a statistical sense. All four methods require similar amounts of computer time.

Avoidance-modified generalised distributions and their application to studies of superparasitism.

Abstract: Given an underlying distribution for the number of encounters between parasites and a host this may be generalised by the distribution of the number of eggs per encounter. If eggs are already present, a parasite may avoid oviposition at subsequent encounters. A class of avoidance-modified generalised distributions is presented to model such situations. The case of a solitary (single egg-laying) parasite and a Poisson distribution for the number of encounters gives rise to a distribution of particular interest. Various properties of this distribution are derived and alternative estimation procedures investigated. Estimators based on the method of mean and zero frequency are shown to have the dual advantages over other estimators that have been proposed of simplicity of computation and increased efficiency. Over a wide range of parameter values there is virtually no loss of efficiency in comparison with maximum likelihood estimation.

Species Abundance with Optimum Relations to Environmental Factors

Abstract: Models are described for the numbers of individuals of species j on site i. It is assumed that the numbers can be conceived as independent trials from Poisson distributions. The expected numbers are thought to be a function of one or two environmental variables. This function is chosen to be Gaussian. Statistical tests are presented for goodness of fit and for contrasting several hypotheses concerning these models. Listings of computer programs for estimating the parameters involved, are available on request. A comparison of the models with the principal component analysis is included.

The estimation of a probability density function from measurements corrupted by Poisson noise (Corresp.)

Abstract: The estimation of a probability density function from measurements corrupted by independent additive Poisson noise is considered. An estimate is derived that is asymptotically unbiased and consistent in the quadratic mean. Also, a practical realization of the estimator is given.

Planning the suicide experiments.

Abstract: Suicide experiments involve a great degree of uncertainty in the counting of cell colonies. This work studies from a statistical point of view the precision of the estimation as a function of the number of experimental units which are used. Assuming that the colony numbers in recipients follow a Poisson distribution, we give the necessary number of recipients (a) to determine with a given accuracy the percentage of DNA synthesizing cells (S cells), (b) to test whether or not a cell population is quiescent, and (c) to compare the percentages of S cells in two cell populations.

Renewal 'theory in two dimensions: bounds on the renewal function

Abstract: In two earlier papers [6], [7] the properties of bivariate renewal processes and their associated two-dimensional renewal functions, H(x, y) were examined. By utilising the Fr6chet bounds for joint distributions and the properties of univariate renewal processes, a collection of upper and lower bounds for H(x, y) are constructed. The evaluation of these bounds is carried out for the case of the family of bivariate Poisson processes. An interesting by-product of this investigation leads to a new inequality for the median of a Poisson random variable.

Two Applications of a Poisson Approximation for Dependent Events

Abstract: Recent results have estimated the error when sums of dependent nonnegative integer-valued random variables are approximated in distribution by a Poisson variable. Two problems are considered where these results can be used to provide simple solutions. The first problem studies the asymptotic behavior, as $\alpha \rightarrow 0$, of the number of independent random arcs of length $\alpha$ needed to cover a circle of unit circumference at least $m$ times $(m \geqq 1)$. The second problem deals with urn schemes.

A continuous review inventory model with compound poisson demand process and stochastic lead time

Abstract: Using Markov renewal theory, we derive analytic expressions for the expected average cost associated with (s, S) policies for a continuous review inventory model with a compound Poisson demand process and stochastic lead time, under the (restrictive) assumption that only one order can be outstanding.

Wiener analysis of nonlinear systems using Poisson-Charlier crosscorrelation

Abstract: Nonlinear systems with event-sequence input, such as are often encountered in neurophysiology, may be experimentally tested with all possible input sequences by stimulation with a Poisson process eventsequence. A complete predictive model of the system's response may be constructed from this data with the Wiener expansion based on the Poisson-Charlier polynomials. Here it is shown how this formulation leads to an efficient method for the evaluation of unknown systems by crosscorrelation, generalizing previous methods. The basic statistical properties of the procedure are demonstrated and the length of experiment required for accurate estimation of the model is computed. The procedure is translated into digital algorithms and the analogous procedures for white noise analysis are presented.