Showing papers on "Probability-generating function published in 1981"

Self-decomposable discrete distributions and branching processes

Abstract: Self-decomposable distributions are known to be absolutely continuous. In this note analogues of the concept of self-decomposability are proposed for distributions on the set ℕ0 of nonnegative integers. To each of them corresponds an analogue of multiplication (in distribution) that preserves ℕ0-valuedness and is characterized by a composition semigroup of probability generating functions, such as occur in branching processes.

On the computer generation of random variables with a given characteristic function

Abstract: We consider the problem of the computer generation of a random variable X with a given characteristic function when the corresponding density and distribution function are not explicitly known or have complicated explicit formulas. Under mild conditions on the characteristic function, we propose and analyze a rejection/squeeze algorithm which requires the evaluation of one integral at a crucial stage.

Some results for the general bulk service queueing system

Abstract: This paper deals with a general bulk service queueing system with one server, for which customers arrive in a Poisson stream and the service is in bulk. The maximum number of customers to be served in one lot is B (capacity), but the server does not start service until A (quorum, less than B) customers have accumulated; and the service time follows a general probability distribution. Probability generating functions of the distributions Pn and , under equilibrium, have been derived by using the supplementary variable technique. An expression has been derived for the expected value of the queue size and its relation with the expected value of the waiting time of a customer has been explored. A numerical case has been worked out on the assumption that bulk service follows a specified Erlangian distribution.

On a Class of Estimates of the Probability Density Function and mode based on a Random Number of Observations

Abstract: The estimate of the probability density function, based on a fixed number of observations, studied by Yamato (1971) and Davies (1973), has been extended to the case when the number of observations is random. Asymptotic properties of the estimates of She d:osity function and its derivatives, as also of the estimate of the mode, have been studied under appropriate conditions.

On Bounds for Probability Generating Functions

Abstract: Summary Brook (1966) gave an upper bound for the moment generating function (m.g.f.) of a positive random variable (r.v.) in terms of its moments, and used this to obtain an upper bound for the probability generating function (p.g.f.) and hence the extinction probability of a simple branching process. Agresti (1974) rederived this bound of the p.g.f. and used it to obtain a lower bound of the expectation of extinction time of a branching process. In both of these applications the random variable is integer valued, and for this class we improve on Brook's bound by deriving the best upper bound of the p.g.f. Our method, which is a variant of Brook's (1966) is used later to obtain the lower bound of the p.g.f. when the third moment is also known.

Relation of modified power series distributions to lagrangian probability distributions

Abstract: The class of Lagrangian probability distributions ‘LPD’, given by the expansion of a probability generating function f‘t’ under the transformation u = t/g‘t’ where g ‘t’ is also a p.g.f., has been substantially widened by removing the restriction that the defining functions g ‘t’ and f‘t’ be probability generating functions. The class of modified power series distributions defined by Gupta ‘1974’ has been shown to be a sub-class of the wider class of LPDs

Bivariate Generalized Discrete Distributions and Bipartitional Polynomials

Abstract: If Z and (X, Y) are independent discrete random variables with probability generating functions f(u) and g(τ, t), respectively, then the generalized random variable (V,W) has probability generating function h (τ, t) = f (g (τ, t)). This class of bivariate discrete distributions includes many of the known bivariate contagious and compound distributions. In the present paper the probabilities and factorial moments of (V,W) are expressed in terms of the probabilities and factorial moments of Z and (X ,Y) through the bipartitional polynomials; these polynomials are multivariable polynomials Ymn(fg01, fg10, fg11, ..., fgmn), fk≡fk defined by a sum over all partitions of their bipartite indices (m , n). Using properties of these polynomials, the conditional probabilities and factorial moments of W given V=m are obtained in explicit forms. Recurrence relations of the probabilities and factorial moments are obtained by using the general recurrence relation for the bipartitional polynomials. These general results are applied to the bivariate generalized Poisson, logarithmic series and general binomial distributions. Moreover, certain bivariate generalized discrete distributions with specified the generalizing random vector (X, Y) are briefly discussed.

Deriving Sediment Yield Probabilities for Evaluating Conservation Practices

Abstract: PROBABILITY distributions of annual sediment yield are derived for both conventional and minimum tillage of watershed cropland. For the derivation, a recursive technique derived from probability generating functions is used to couple a bivariate rainfall model and a stochastic watershed state model to a deterministic model of the sediment yield process. The bivariate rain-fall model consists of a Weibull PDF for rainfall event duration and a log-normal PDF for event depth, given duration; the stochastic state model is a probability mass function for antecedent rainfall class; the deterministic model is composed of the Williams sediment-yield model and SCS techniques for computing direct runoff volume and peak runoff rate.

Fixed and random coefficient time series

Abstract: Linear modelling of data has been a major concern in time series for many years. One of the last outstanding problems has been the determination of the order of a model once a class of models has been decided upon. In Chapter 2 automatic criteria for the determination of the order of a multivariate autoregression are discussed. Via a law of the iterated logarithm for martingales, it is shown under what conditions an automatic criterion of a certain type is strongly consistent for the true order of the autoregression.