Showing papers on "Poisson distribution published in 1973"
TL;DR: In this paper, a generalization of the Poisson distribution with two parameters λ1 and λ2 is obtained as a limiting form of the generalized negative binomial distribution, where the variance of the distribution is greater than, equal to or smaller than the mean according as λ 2 is positive, zero or negative.
Abstract: A new generalization of the Poisson distribution, with two parameters λ1 and λ2, is obtained as a limiting form of the generalized negative binomial distribution. The variance of the distribution is greater than, equal to or smaller than the mean according as λ2 is positive, zero or negative. The distribution gives a very close fit to supposedly binomial, Poisson and negative-binomial data and provides with a model suitable to most unimodel or reverse J-shaped distributions. Diagrams showing the variations in the form of the distribution for different values of λ1 and λ2 are given.
518 citations
TL;DR: Numerical investigation of errors in the approximation and subsequent experience has shown that this method of generating overflow traffic is accurate and very useful in both simulations and analyses of traffic systems.
Abstract: Traffic overflowing a first-choice trunk group can be approximated accurately by a simple renewal process called an interrupted Poisson process–a Poisson process which is alternately turned on for an exponentially distributed time and then turned off for another (independent) exponentially distributed time. The approximation is obtained by matching either the first two or three moments of an interrupted Poisson process to those of an overflow process. Numerical investigation of errors in the approximation and subsequent experience has shown that this method of generating overflow traffic is accurate and very useful in both simulations and analyses of traffic systems.
264 citations
TL;DR: In this paper, the principle of maximum likelihood is used to obtain estimates of the parameters in a regression model when the experimental observations are assumed to follow the Poisson distribution, and the maximum likelihood estimates are shown to be equivalent to those obtained by minimization of a quadratic form which reduces to a modified chi square under the assumption.
Abstract: The principle of maximum likelihood is used to obtain estimates of the parameters in a regression model when the experimental observations are assumed to follow the Poisson distribution. The maximum likelihood estimates are shown to be equivalent to those obtained by minimization of a quadratic form which reduces to a modified chi square under the Poisson assumption. Computationally, both of these estimation procedures are equivalent to a properly weighted least squares analysis. Approximate tests of the assumed Poisson variation and “goodness of fit” of the data to the model are proposed. Applications of the estimation procedure to linear and nonlinear regression models are discussed, and numerical examples are presented.
162 citations
Book•
01 Jan 1973
TL;DR: Theorems of Bernoulli and Stirling and the binomial, Poisson and hypergeometric distributions are cited as sources of uncertainty for the Gaussian distribution.
Abstract: Uncertainties and frequency distributions The Gaussian distribution General distributions Rectangular distributions Applications Distributions ancillary to the Gaussian A general theory of uncertainty The estimation of calibration uncertainties Consistency and significance tests Method of least squares Theorems of Bernoulli and Stirling and the binomial, Poisson and hypergeometric distributions Appendices Bibliography Index
110 citations
TL;DR: In this paper, the authors deal with methods of quality control when observation-vectors are coming from a multivariate binomial or multivariate Poisson population, under certain assumptions.
Abstract: This paper deals with methods of quality control when observation-vectors are coming from a multivariate binomial or multivariate Poisson population. The case when successive observations are time-dependent has been studied under certain assumptions. Further when a sample covariance matrix is singular or near singular, a method has been proposed for translating it into a non-singular estimate under the assumption of conditional independence. Assuming approximate normality a x 2 -chart has been proposed. At the end a numerical example is given to illustrate the procedure in this paper.
100 citations
TL;DR: This article showed that for a sequence of dependent events, such that each has uniformly small conditional probability given the past, and the sum of the conditional probabilities is nearly constant at $a, the number of events which occur is nearly Poisson with parameter $a.
Abstract: Here is another way to prove Levy's conditional form of the Borel-Cantelli lemmas, and his strong law. Consider a sequence of dependent variables, each bounded between 0 and 1. Then the sum $S$ of the variables tends to be close to the sum $T$ of the conditional expectations. Indeed, the chance that $S$ is above one level and $T$ is below another is exponentially small. So is the chance that $S$ is below one level and $T$ is above another. The inequalities also show that for a sequence of dependent events, such that each has uniformly small conditional probability given the past, and the sum of the conditional probabilities is nearly constant at $a$, the number of events which occur is nearly Poisson with parameter $a$.
69 citations
TL;DR: In this article, the probability distributions of various aggregates of random convex polygons determined by the standard isotropic Poisson line process in the plane are investigated, defined in an ergodic sense.
66 citations
TL;DR: In this article, it was shown that the number of jumps of an extremal Markov process in (tl, t2] follows a Poisson distribution with parameter log t2/t1.
Abstract: An extremal-F process { Y (t); t > 0} is defined as the continuous time analogue of sample sequences of maxima of i.i.d. r.v.'s distributed like F in the same way that processes with stationary independent increments (s.i.i.) are the continuous time analogue of sample sums of i.i.d. r.v.'s with an infinitely divisible distribution. Extremal-F processes are stochastically continuous Markov jump processes which traverse the interval of concentration of F. Most extremal processes of interest are broad sense equivalent to the largest positive jump of a suitable s.i.i. process and this together with known results from the theory of record values enables one to conclude that the number of jumps of Y (t) in (tl, t2] follows a Poisson distribution with parameter log t2/t1. The time transformation t---> et gives a new jump process whose jumps occur according to a homogeneous Poisson process of rate 1. This fact leads to information about the jump times and the interjump times. When F is an extreme value distribution the Y-process has special properties. The most important is that if F(x) = exp {-e-X} then Y(t) has an additive structure. This structure plus non parametric techniques permit a variety of conclusions about the limiting behaviour of Y(t) and its jump times.
58 citations
TL;DR: In this article, the problem of testing the existence of a trend in the means Gi of Poisson distributions is considered. But it is assumed that these means are changing exponentially, that is, log Gi = ci+/x2.
Abstract: SUMMARY This paper is concerned with the problem of testing the existence of a trend in the means Gi of Poisson distributions. It is assumed that these means are changing exponentially, that is, log Gi = ci+/x2. A classical method is reviewed which is used for testing the hypothesis P = 0. The exact Bayesian distribution for P is derived and a Bayesian approximation suggested which proved to be very useful. Finally, a comparison of these three methods by means of numerical examples is made.
50 citations
TL;DR: In this paper, a procedure for estimating the zero class from a truncated Poisson sample is developed, and asymptotic normality of the estimator is proved so that a confidence interval for the missing zero class can be obtained.
Abstract: A procedure for estimating the zero class from a truncated Poisson sample is developed. Asymptotic normality of the estimator is proved so that a confidence interval for the missing zero class can be obtained. An example is given to illustrate the results obtained.
47 citations
TL;DR: The matrix decomposition Poisson solver is developed for the five-point difference approximation to Poisson's equation on a rectangle and its suitability for parallel computation, its simplicity, and its performance relative to successive overrelaxation are discussed.
Abstract: The matrix decomposition Poisson solver is developed for the five-point difference approximation to Poisson's equation on a rectangle This algorithm's suitability for parallel computation, its simplicity, its performance relative to successive overrelaxation, and its generality are then discussed
TL;DR: In this paper, the equivalence of the two types of test ceases to hold when the null hypothesis involves more than one parameter and at least one of these lies on the boundary of the parameter space, contrary to the assumptions under which the C(ca) test is derived.
Abstract: SUMMARY Tests for the homogeneity of samples from the Poisson and Gamma distributions are considered based on the C(ac) procedure of Neyman and on maximum likelihood estimators. These are shown to be equivalent in spite of the fact that the null hypothesis lies on the boundary of the parameter space, which is contrary to the assumptions under which the C(ca) test is derived. Moreover, it is shown that neither procedure has been proved to work unless the disturbing distribution is assumed to have a zero third moment, a so far unexplained phenomenon. Furthermore, it is pointed out that the equivalence of the two types of test ceases to hold when the null hypothesis involves more than one parameter and at least one of these lies on the boundary of the parameter space.
Journal Article•
TL;DR: In this article, the valuation of diamondiferous deposits depends on the density distribution, the size distribution of stones and the selling price structure prevailing at a particular time, and confidence limits are calculated which allow for inferences from sampling to recovery.
Abstract: The valuation of diamondiferous deposits depends on (i) the density distribution, (ii) the size distribution of stones and (iii) the selling price structure prevailing at a particular time. In the density distribution the stochastic variable is defined as the number of individual stones found per sample unit irrespective of the stone size. The density distribution is discrete and allows for barren sample units. As diamonds often occur in clusters, a Poisson mixture distribution is derived which is entirely new, as existing discrete distributions cannot represent the stone number frequencies found in nature. For the sizes of diamonds, which are measured by weight, a new compound logarithmic normal distribution is developed. Combination of (i), (il) and (iii) leads to a meaningful appraisal of diamondiferous ground expressed in monetary value per production unit of mining. Finally, confidence limits are calculated which allow for inferences from sampling to recovery.
01 Dec 1973
TL;DR: In this paper, a modification to the standard test for trend, both for modulated renewal and general point processes, is presented. But the test is not robust with respect to the distribution theory of the underlying point process.
Abstract: : In examining point processes which are overdispersed with respect to a Poisson process, there is a problem of discriminating between trends and the appearance in data of sequences of very long intervals. In this case the standard robust methods for trend analysis based on log transforms and regression techniques perform very poorly, and the standard exact test for a monotone trend derived for modulated Poisson processes is not robust with respect to its distribution theory when the underlying process is non-Poisson. However, experience with data and an examination of the departures from the Poisson distribution theory suggest a modification to the standard test for trend, both for modulated renewal and general point processes.
TL;DR: In this paper, the conditional distribution of the state of the Markov chain, given observations of the point process, is derived for the forward recurrence time of a point process and distributions corresponding to synchronous and asynchronous sampling are also derived.
Abstract: For a continuous time Markov chain the time points of transitions, belonging to a subset of the set of all transitions, are observed. Special cases include the point process generated by all transitions and doubly stochastic Poisson processes with a Markovian intensity. Equations are derived for the conditional distribution of the state of the Markov chain, given observations of the point process. This distribution may be used for prediction. For the forward recurrence time of the point process, distributions corresponding to synchronous and asynchronous sampling are also derived. The Palm distribution for the point process is specified in terms of the corresponding initial distribution for the Markov chain. In examples the point processes of arrivals and departures in a queueing system are studied. Two biological applications deal with estimation of population size and detection of epidemics. POINT PROCESS; MARKOV CHAIN; STATE ESTIMATION; PREDICTION OF POINT PROCESSES; PALM PROBABILITIES; DOUBLY STOCHASTIC POISSON PROCESSES; ESTIMATION OF POPULATION SIZE; DETECTION OF EPIDEMICS
TL;DR: In this paper, the authors investigated the variation in point precipitation with elevation using an event-based stochastic model of thunderstorm rainfall and empirical data and used the model to obtain the return period of storm rainfall of a given magnitude despite a short historical record.
TL;DR: In this article, the authors studied the decay of an initial Poisson distribution (coherent state) to the equilibrium Bose-Einstein distribution in a mixed Poisson process.
Abstract: The master equation formed from the diagonal elements of the density matrix is solved in general for arbitrary initial photon probability distributions. Several examples are studied including the decay of an initial Poisson distribution (coherent state) to the equilibrium Bose-Einstein distribution. The concept of a mixed Poisson process is introduced and its physical implications examined in the context of the present problem. A general expression for the nonstationary correlation function of the photon field is also obtained.
TL;DR: In this paper, a Poisson limit is established for the probability that $k$ events occur and an asymptotic distribution for the number of upcrossings of a high level by certain stochastic processes is considered as an application.
Abstract: Certain mixing sequences of dependent `rare' events are considered and a Poisson limit is established for the probability that $k$ events occur. An asymptotic distribution for the number of upcrossings of a high level by certain stochastic processes is considered as an application.
TL;DR: In this paper, the exact distribution of the test-statistic u-test was derived and the approach to normality was shown that the normality is slow, and a practical question dealt with the extrapolation of Ractliffe's results to Poisson expectations with A < 5.
Abstract: A further practical question deals with the extrapolation of Ractliffe's results to Poisson expectations with A < 5. In this paper, the exact distribution of the test-statistic u will be derived. It will be shown that the approach to normality is slow. Formulae will be given for numerical calculation of individual and cumulative probabilities. Particularly for a probability level of 5 per cent (double-tail), the u-test performs much better than the customary "binomial" test, with the added advantage of simplicity and non-requirement of tabulated values. The u-test may safely be applied from A = 2 5 upwards. 50
TL;DR: It is found that the optimal strategy is a somewhat complex but very tractable function of the two Poisson arrival parameters.
Abstract: A vehicle shuttles between two terminals, at which passengers are arriving in a Poisson manner at given rates. Travel time between the stops is assumed deterministic. This paper considers the question of what operating strategy will yield lowest average waiting time for passengers of the system. Exact solutions to the problem are obtained, and it is found that the optimal strategy is a somewhat complex but very tractable function of the two Poisson arrival parameters.
TL;DR: In this article, the dependence of the spectral condition number of the finite element stiffness matrix, in the case of a nearly incompressible solid, was established as a function of the mesh size and Poisson's ratio.
TL;DR: The equilibrium virtual delay distribution is derived and compared with the equilibrium distribution of delays seen by arriving calls, showing that delays can be quite sensitive to peakedness.
Abstract: This paper treats the problem of analyzing a first-come first-served queuing system, in equilibrium, when subjected to a peaked input (e.g., traffic overflowing a trunk group with Poisson input). The basic GI/M/N (renewal input to N exponential servers) queuing result is used, together with each of two models for representing peaked traffic, the Equivalent Random (E-R) model and the Interrupted Poisson Process (IPP) model. The equilibrium virtual delay distribution is derived and compared with the equilibrium distribution of delays seen by arriving calls. Numerical examples are presented, along with comparisons of results using both the above models. The results show that delays can be quite sensitive to peakedness.
TL;DR: In this paper, a chain of inequalities for probabilities in some types of multivariate distributions is proved, and the inequalities embrace a large number of interesting special cases, such as normal and rank statistics for comparing many treatments with one control, order statistics used in estimating quantiles, and characteristic roots of covariance matrices in certain multiple sampling.
Abstract: Generalizing the result by Y. L. Tong, a chain of inequalities for probabilities in some types of multivariate distributions is proved. These inequalities embrace a large number of interesting special cases. Nine illustrations are given: cases of multivariate equicorrelated normal, $t,\chi^2$, Poisson, exponential distributions, normal and rank statistics for comparing many treatments with one control, order statistics used in estimating quantiles, and characteristic roots of covariance matrices in certain multiple sampling.