Showing papers on "Poisson distribution published in 1974"

A cluster process representation of a self-exciting process

Abstract: It is shown that all stationary self-exciting point processes with finite intensity may be represented as Poisson cluster processes which are age-dependent immigration-birth processes, and their existence is established. This result is used to derive some counting and interval properties of these processes using the probability generating functional.

Computer methods for sampling from gamma, beta, poisson and bionomial distributions

Abstract: Accurate computer methods are evaluated which transform uniformly distributed random numbers into quantities that follow gamma, beta, Poisson, binomial and negative-binomial distributions. All algorithms are designed for variable parameters. The known convenient methods are slow when the parameters are large. Therefore new procedures are introduced which can cope efficiently with parameters of all sizes. Some algorithms require sampling from the normal distribution as an intermediate step. In the reported computer experiments the normal deviates were obtained from a recent method which is also described.

On Fitting the Poisson Lognormal Distribution to Species-Abundance Data

Abstract: An extension of MacArthur's "broken stick" model is proposed to explain why species abundances should be lognormally distributed. A method of fitting the compound Poisson lognormal distribution by maximum likelihood is described; a computer program is available for performing the calculations. It is shown how the information theory measure of species diversity can be estimated from the parameters of the fitted distribution.

Buffer Behavior with Poisson Arrival and Geometric Output Processes

Abstract: A discrete buffered system with infinite buffer size, Poisson arrival process, and the output channel available only periodically according to a geometric density function is analyzed. Under the assumption that a stochastic equilibrium is reached, the Z transform of the density function of the buffer occupancy is obtained as the result of this study.

Log-Linear Models for Frequency Tables Derived by Indirect Observation: Maximum Likelihood Equations

Abstract: Frequency tables are examined in which some cells are not distinguishable. Log-linear models are proposed for these tables which lead to likelihood equations closely related to those associated with log-linear models for conventional frequency tables. Just as in conventional tables, the maximum likelihood equations are shown to be the same under Poisson or multinomial sampling. Applications are made to the problem of estimation of gene frequencies from observed phenotype frequencies.

On the Convergence of Poisson Binomial to Poisson Distributions

Abstract: It is well known that under certain conditions, binomial distributions converge to Poisson distributions. Simons and Johnson (1971) showed that the convergence is actually much stronger than in the usual sense. In this note the author shows that the result of Simons and Johnson is also true for Poisson binomial distributions which include binomial distributions as special cases.

Bounds on the extinction time distributions of some branching processes

Abstract: The class of fractional linear generating functions, one of the few known classes of probability generating functions whose iterates can be explicitly stated, is examined. The method of bounding a probability generating function g (satisfying g ″(1) g . For the special case of the Poisson probability generating function, the best possible bounding fractional linear generating functions are obtained, and the bounds for the expected time to extinction of the corresponding Poisson branching process are better than any previously published.

Statistical-Physical Models of Man-Made Radio Noise, Part I. First-Order Probability Models of the Instantaneous Amplitude

Abstract: A general statistical-physical model of man-made radio noise processes appearing in the input stages of a typical receiver is described analytically. The first-order statistics of the se random processes are developed in detail for narrow-band reception. These include, principally, the first order probability densities and probability distributions for a) a purely impulsive (poisson) process, and b) an additive mixture of a gauss background noise and impulsive sources. Particular attention is given to the basic waveforms of the emissions. in the course of propagation. including such critical geometric and kinematic factors as the beam patterns of source and receiver, mutual location, Doppler, far-field conditions, and the physical density of the sources, which are assumed independent and poisson distributed in space over a domain A. Apart from specific analytic relations. the most important general result s are that these first-order distributions are analytically tractable and canonical. They are not so complex as to be unusable in communication theory applications; they incorporate in an explicit way the controlling physical parameters and mechanisms which determine the actual radiated and received processes; and finally, they are formally invariant of the particular source location and density, waveform emission, propagation mode, etc., as long as the received disturbance is narrow-band, at least as it is passed by the initial stages of the typical receiver. The desired first-order distributions are represented by an asymptotic development, with additional terms dependent on the fourth and higher moments of the basic interference waveform, which in turn progressively affect the behavior at the larger amplitudes. This first report constitutes an initial step in a program to provide workable analytical models of the general nongaussian channel ubiquitous in practical communications applications. Specifically treated here are the important classes of interference with bandwidths comparable to (or less than) the effective aperture-RF-IF bandwidth of the receiver, the common situation in the case of communication interference.

The stochastic properties of spontaneous quantal release of transmitter at the frog neuromuscular junction

Abstract: 1. Earlier results showed that it is unlikely that spontaneous quantal release of transmitter at the frog neuromuscular junction is produced by a Poisson process. 2. Data sets were tested, by using the u statistic, to see whether if they are assumed to be generated by a Poisson process, the mean interval is changing monotonically with time. By this critieria, some of the data sets are stationary, others are not. 3. A variety of mathematical transforms are employed on empirical data sets to characterize the properties of the spontaneous quantal release. (a) The intensity function, which calculates the frequency distribution of all possible combinations of intervals, shows an excess of short intervals, without any sign of periodicity. (b) The variance—time curve, which estimates the accumulated variance of the series as a function of time into the series, lies significantly above the Poisson prediction. (c) The power spectrum, whether calculated on the intervals or on the number of intervals in time bins, deviates significantly from the Poisson prediction at the low frequencies. (d) The ln-survivor curve has two phases: a concave section for the short intervals, and a roughly linear section for the intervals of greater length. These transforms indicate that the min.e.p.p.s are clustered. 4. A series of models for spontaneous quantal release were considered. (a) A Poisson model. Rejected because of consistent failure to fit the data. (b) A periodic model. Rejected because the intervals should be ordered rather than clustered. (c) A time-dependent model, in which quantal release is governed by a Poisson process with a mean interval that is oscillating in time. This model will generate clustering; by the transforms the model can be shown to closely fit the data. However, an autocorrelation of min.e.p.p. amplitudes shows that there is a relationship between the amplitudes and their position in the series. This is not predicted by the time-dependent oscillating model. (d) A branching Poisson model, in which a primary release, generated by a Poisson process, is likely to be followed by one or more subsidiary releases from the same site. The parameters of the branching model can be determined from ln-survivor curves. Theoretical curves, created with these parameters, give power spectra, variance—time curves, and ln-survivor curves that strongly resemble those calculated from the data. The model also predicts a significant autocorrelation of amplitudes. 5. Min.e.p.p.s recorded with an extracellular electrode also fit well to a branching Poisson model. 6. The effects of raised [Ca2+]o on the intervals between min.e.p.p.s were studied. In our experiments the change in extracellular solution did not produce any notable change in release statistics. 7. The effects of elevated [K+]o on the intervals between spontaneous releases were studied. Depolarization of the nerve terminal increases the frequency of primary releases and decreases the chance of having subsidiary releases. 8. Possible physical mechanisms by which quantal release of transmitter from a nerve terminal would fit a branching Poisson model are described.

Direct-Detection Optical Communication Receivers

Abstract: A model that is sufficiently general to describe the predominant statistical characteristics of the output of many real optical detectors is formulated. This model is used to study the optimum receiver processing for direct-detection optical communication systems. In particular, the structures of detectors and estimators for randomly filtered doubly stochastic Poisson processes observed in additive white Gaussian noise are considered. Representations for the posterior statistics of a vector-valued Markov process that modulates the intensity of the doubly stochastic Poisson process are obtained. Quasi-optimum estimators and detectors are specified in general terms and specialized for several important applications. These include a demodulator for subcarrier angle modulation, a detector structure for binary signaling with known intensities, and a detector structure for binary signaling in the turbulent atmosphere.

The intervals between miniature end‐plate potentials in the frog are unlikely to be independently or exponentially distributed

Abstract: 1. It has been suggested that spontaneous quantal release of transmitter at the neuromuscular junction is a Poisson process. One logical argument against accepting the Poisson hypothesis is that so far relatively few intervals between miniature end-plate potentials (min.e.p.p.s) have been studied in any single experiment. Release is known to occur from many sites on the nerve terminal, so many intervals must be studied before drawing any conclusions about the timing of release from the individual sites. Moreover, the statistical methods that have been used are relatively insensitive to deviations from Poisson predictions.2. The Poisson hypothesis is evaluated with respect to three major criteria:(a) The fit to the exponential distribution is analysed by five goodness of fit tests which were applied to eleven sets of data, showing that it is unlikely that the data sets were generated by an exponential distribution.(b) The independence of intervals is assessed in two ways. First, the autocorrelogram of intervals is constructed. This shows an excess of significant positive correlations beyond the 5% limits of the Poisson expectation. Secondly, the unsmoothed power spectrum is calculated, and compared to the Poisson prediction by means of the modified mean test. Again, most sets deviate significantly from the Poisson expectation. It is unlikely that the intervals are independent.(c) The possibility of simultaneous occurrences is evaluated by construction of the amplitude histogram of min.e.p.p.s. In all sets the Poisson prediction for the frequency of multiples of the unit height was exceeded by the empirical data sets. The over-all conclusion is that the process which generates spontaneous releases is unlikely to be Poisson.

The poisson approximation for dependent events

Abstract: Consider a sequence of dependent events, where each has uniformly small conditional probability given the past, and the sum of the conditional probabilities is approximately constant at $a$. Then the number of events which occur is approximately Poisson with parameter $a$. An explicit bound is given on the variation distance.

Canonical ratings

Abstract: The concept of canonical ratings is introduced in which each S describes all the visual sensations produced by signal and noise trials in the expected spatial and temporal location of the stimulus. After many practice sessions, the S assigns one and only one numerical rating to each visual sensation. These canonical ratings are determined by the S, not the E, and are a, bbreviations for verbal descriptions of subjectively distinct visual sensations. The data consisted of canonical ratings at absolute visual detection for dim visual stimuli (signal) and blank (noise) trials containing no light at all. The physical stimulus is discrete since it is made up of absorptions of quanta of light that result in isomerizations of rhodopsin molecules or thermal decompositions of rhodopsin which are discrete noise events that mimic the action of quantal absorptions. Under these conditions, it is known from the laws of physics that these quantum-like events labsorptions plus thermal decompositions) follow a Poisson distribution. Previously, it had been shown that the canonical ratings follow the same Poisson distributions that the quantum-like events do. It was also shown that the data for one S were consistent with the hypothesis that the rating on any trial was equal to the number of quantum-like events that had occurred and for two other Ss, either one less or two less than this number. A signal detection theory analysis of these canonical ratings is performed, resulting in ROC curves and estimates of d’. In addition, it is shown that the Poisson canonical rating distributions can be approximated by cutoff Gaussian distributions. Hence it is possible to use a probit analysis, which is computationally simple, to calculate the maximum likelihood solutions for all means, standard deviations, d’, and b, as well as the standard errors of all these estimates. The rating is shown to be a linear function of the internal decision variable. The internal criteria are all greater than the mean of the noise distribution and they are all separated by steps of equal size. The probit analysis may be used whenever all the individual rating distributions are Gaussian in order to obtain the maximum likelihood estimates and standard errors of all parameters for each Gaussian distribution. Thus, this analysis may be applied to rating experiments, other than the one described here.

Probabilistic Models for Texas Gulf Coast Hurricane Occurrences

Abstract: Various statistical methods are used to analyze the occurrence of Texas Gulf Coast hurricanes. Simple Poisson, periodic Poisson, and Markov chain models are fitted to the occurrence data for an offshore site near Corpus Christi, Texas. The goodness of the fit of the models is studied, using interarrival time, hazard function, and comparative maximum likelihood tests. Also considered are the autocorrelation function for the data and the effects of varying record lengths. The periodic Poisson model provides the best fit to the data and permits a relatively simple description of cyclical occurence phenomena. (AUTHOR)

Information rates and data-compression schemes for Poisson processes

Abstract: In this paper, we derive rate-distortion functions under proper magnitude-error fidelity criteria and study instrumentable data-compression schemes for Poisson processes. In particular, we derive information rates and obtain rate-distortion relationships for practical data-compression schemes, for the reproduction of the unordered sequence of Poisson event occurrences, for the reproduction of the sample functions of the Poisson counting process, and for the reproduction of the sequence of intervals between the event occurrences of a Poisson process. The reproducing processes are taken to be point (or jump) processes themselves. The performances of the various data-compression schemes presented here are compared with those of the ideal schemes (us presented by the rate-distortion functions) and are shown to be close to the latter over wide regions of distortion.

A characterization of the Poisson process

Abstract: Theorem: A necessary and sufficient condition for the superposition of two ordinary renewal processes to again be a renewal process is that they be Poisson processes. A complete proof of this theorem is given; also it is shown how the theorem follows from the corresponding one for the superposition of two stationary renewal processes.

Alternative Process Models in the Economic Design of T2 Control Charts

Abstract: The use of three probability distributions to model the time to failure in an economic model of a multivariate quality control procedure is investigated. A discrete-time Markov (geometric) model is developed, as well as two non-Markovian (Poisson and logarithmic series) models. Numerical examples are presented which indicate that both the Markov assumption and the shape of the distribution of time to failure are of considerable importance in determining the optimal test parameters.

A finite dam with exponential release

Abstract: This paper considers a finite dam with independently and identically distributed (i.i.d.) inputs occurring in a Poisson process; the special cases where the inputs are (i) deterministic and (ii) negative exponentially distributed are considered in detail. The instantaneous release trate is proportional to the content, i.e., there is an exponential fall in conten except when inputs occur. This model may arise in several other situations such as a geiger counter or integrated shot noise. The distribution of the number of inputs, and of the time, to first overflowing is obtained in terms of generating functions; in Case (i) the solution is obtained through recurrence relations involving iterated integrals which can be evaluated numerically, and in Case (ii) using a series solution of a second order differential equation. Numerical results, in particular for the first two moments, are obtained for various values of the parameters of the model, and compared with a large number of simulations. Some remarks are also made about the infinite dam. FINITE DAMS; POISSON INPUTS; EXPONENTIAL RELEASE; FIRST PASSAGE TIMES; RECURRENCE RELATIONS, NUMERICAL INTEGRATION; SERIES SOLUTION; SIMULATION

The Estimation of Population Size with Equal and Unequal Risks of Capture

Abstract: Behavioral characteristics and microhabitat distributions of individuals may cause heterogeneity in their risk of capture in field studies of the dynamics of natural populations. The number of recaptures of marked individuals can be used to test for equicatchability by their fit to a Poisson distribution. If the null hypothesis of equal risk of capture is rejected the number of captures per individual may follow a negative binomial distribution. The use of truncated distributions to estimate the number of transients in the population, or the number of animals never caught is found to be of very limited value. These methods are applied to an original set of capture histories from a study of a population of the painted turtle, Chrysemys picta. See full-text article at JSTOR

An elementary proof for the rao-rubin characterization of the poisson distribution

Abstract: This note gives elementary proofs for the characterizations of the Poisson distribution given by Rao and Rubin (1964) and Talwalker (1970) and disproves a conjecture of R. C. and A. B. L. Srivastava (1970). CHARACTERIZATION, POISSON DISTRIBUTION

A Queueing Problem in Which Customers Have Different Service Distributions

Abstract: A single‐server queue is examined whose customers can be categorized into k groups. For the ith group : (1) Arrivals follow a Poisson distribution, the average arrival rate per unit time being λi. (2) The service time follows the exponential distribution with average service time equal to 1/μi. Using a method of “cuts”, the recurrence relations between states of the queueing system are examined. The equilibrium probabilities of queue size are found and it is shown that the approximation of a group of exponential service distributions by a single exponential distribution, having a mean equal to that of the group, always underestimates the mean queue size.