Showing papers on "Probability-generating function published in 1975"

Estimation of Parameters on Some Extensions of the Katz Family of Discrete Distributions Involving Hypergeometric Functions

Abstract: A two-parameter family of discrete distributions developed by Katz (1963) is extended to three- and four-parameter families whose probability generating functions involve hypergeometric functions. This extension contains other distributions appearing in the literature as particular cases. Various methods of estimating the parameters are investigated and their asymptotic efficiency relative to maximum likelihood estimators compared.

Non-parametric estimation in the Galton-Watson process

Abstract: Estimators for the offspring probabilities and probability generating functions and for the extinction probability are defined for the Galton-Watson branching process. These estimators are shown to be conditionally consistent and asymptotically normal in the non-subcritical case and to lack these properties in the subcritical case. Corresponding questions are considered when immigration is permitted.

Use of power transformations to model the yield of IC's as a function of active circuit area

Abstract: Various attempts have been made to analyze the yield of IC's as a function of active circuit area, using as models different probability distribution functions of spot defects. The purpose of this paper is to show that most of the proposed models form a family of functions which correspond to a parametric family of power transformations from yield to yield to the power λ, the parameter λ defining a particular transformation. A comparison between the power transformation approach to earlier modeling techniques will be discussed. After developing the necessary statistical theory, a simple procedure for the model building process will be presented.

On a continuous/discrete time queueing system with arrivals in batches of variable size and correlated departures

Abstract: This paper studies the behaviour of a first-come-first-served queueing network with arrivals in batches of variable size and a certain service time distribution. The arrivals and departures of customers can take place only at the transition time marks and the intertransition time obeys a general distribution. The Laplace transforms of the probability generating functions for the queue length are obtained in the two cases; (i) when departures are correlated; (ii) when departures are uncorrelated; and the steady state results are derived therefrom. It has also been shown that the steady state continuous time solution is the same as the steady state discrete time solution.

Random power series

Abstract: This chapter discusses a particular kind of series of random variables with power series whose coefficients are random variables and explains the connection between the different modes of convergence of the random power series. In general, no circle of convergence exists for convergence in the probability of a random power series. Every random power series defined on a probability space has a circle of convergence in probability if the probability space is atomic. The chapter discusses the behavior of the radius of convergence. The convergence (or divergence) of a random power series is a tail event. It follows immediately from the zero-one law that a random power series with independent coefficients is either almost certainly convergent or almost certainly divergent. The radius of convergence of a convergent random power series with independently and identically distributed coefficients is almost certainly equal to one.

Polynomial bounds for probability generating functions

Abstract: The problem of approximating an arbitrary probability generating function (p.g.f.) g(s) = 2=0o pjsJ by a polynomial LN(S) -= Io rsN is considered. It is shown that if the coefficients r. are chosen so that L N( ) agrees with g(. ) to k derivatives at s = 1 and to (N- k) derivatives at s = 0, then LN is in fact an upper or lower bound to g; the nature of the bound depends only on k and not on N. Application of the results to the problems of finding bounds for extinction probabilities, extinction time distributions and moments of branching process distributions are examined. PROBABILITY GENERATING FUNCTIONS; BRANCHING PROCESSES; EXTINCTION TIME DISTRIBUTIONS; MOMENT PROBLEM

Probability Models for Time Series

Abstract: This chapter describes a variety of probability models for time series, which are collectively called stochastic processes. Most physical processes in the real world involve a random or stochastic element in their structure, and a stochastic process can be described as ‘a statistical phenomenon that evolves in time according to probabilistic laws’. Well-known examples are the length of a queue, the size of a bacterial colony, and the air temperature on successive days at a particular site. Many authors use the term ‘stochastic process’ to describe both the real physical process and a mathematical model of it. The word ‘stochastic’, which is of Greek origin, is used to mean ‘pertaining to chance’, and many writers use ‘random process’ as a synonym for stochastic process.

A generalized neyman's type a contribution for the busy period of queues

Abstract: This paper consider the probability distribution of the busy period of a single server queuing system with constant servicing time for each customer when customer arrive condonly in varying batch sizes and the batch size is distributed according to the Poisson law. First four moments and a few limiting forms of the distribution are also given.

On a supplementary time variable approach to battle analysis

Abstract: A supplementary time variable approach is used to analyze a battle where attackers search for defended logistical targets and the future actions of an attacker depend on its activities in the whole time interval since arrival at the operating area. As compared with the method discussed in this paper a corresponding Markov chain analysis of the problem requires a considerably greater number of mathematical relations for state transition probabilities with less possibilities in obtaining numerical solutions effectively and with less para-metrical insight. A supplementary time variable is introduced to describe the arrival instants of attackers at the operating area, and is then used to express the probability generating functions of the losses at an arbitrarily selected time instant in terms of a general description of the capabilities of an individual attacker in the time interval since arrival. These capabilities are then related to a selected set of lower level parameters describing a specific battle situation by using a small set of integral relations of the multiple convolution type with solutions expressed as Laplace transforms.