Showing papers on "Probability-generating function published in 1994"

Maximum predictive power and the superposition principle

Abstract: We are looking for a way of combining experimentally determined probabilities that should yield maximum predictive power. This concept is defined as permitting calculation of the accuracy of future experimental results solely on the basis of the number of runs whose data will serve as input for making the prediction. Each probability is transformed to an associated variable whose uncertainty interval depends only on the amount of data and strictly decreases with it. We find that for a probability which is a function of two other probabilities maximum predictive power is achieved when linearly summing their associated variables and transforming back to a probability. This recovers the quantum mechanical superposition principle.

Algebraic analysis of the generating functional for discrete random sets and statistical inference for intensity in the discrete Boolean random-set model

Abstract: We consider binary digital images as realizations of a uniformly bounded discrete random set, a mathematical object that can be defined directly on a finite lattice. In this setting we show that it is possible to move between two equivalent probabilistic model specifications. We formulate a restricted version of the discrete-case analog of a Boolean random-set model, obtain its probability mass function, and use some methods of morphological image analysis to derive tools for its statistical inference.

Analyzing the discrete-time G(G)/Geo/1 queue using complex contour integration

Abstract: In this paper, a discrete-time single-server queueing system with an infinite waiting room, referred to as theG(G)/Geo/1 model, i.e., a system with general interarrival-time distribution, general arrival bulk-size distribution and geometrical service times, is studied. A method of analysis based on integration along contours in the complex plane is presented. Using this technique, analytical expressions are obtained for the probability generating functions of the system contents at various observation epochs and of the delay and waiting time of an arbitrary customer, assuming a first-come-first-served queueing discipline, under the single restriction that the probability generating function for the interarrival-time distribution be rational. Furthermore, treating several special cases we rediscover a number of well-known results, such as Hunter's result for theG/Geo/1 model. Finally, as an illustration of the generality of the analysis, it is applied to the derivation of the waiting time and the delay of the more generalG(G)/G/1 model and the system contents of a multi-server buffer-system with independent arrivals and random output interruptions.

The fractional linear probability generating function in the random environment branching process

Abstract: In a branching process with random environments, the probability of ultimate extinction is a function of the environment sequence, and is therefore a random variable. Explicit results about the distribution of this random variable are difficult to obtain in general. Here we assume independent and identically distributed environments and use the special properties of fractional linear generating functions to derive some explicit distributions, which may be singular or absolutely continuous, depending on the values of certain parameters. We also consider briefly tail behaviour close to 1, and provide an extension to cases where probability generating functions are not fractional linear.

Multiple roots in some equations of queueing theory

Abstract: Examples are given of probability generating functions K(z) for which has nonhero multiple roots in 1. Cases in which the roots can be proved to be simple are also discussed

On power series, bell polynomials, hardy-ramanujan-rademacher problem and its statistical applications

Abstract: A possibly unknown approach to the problem of finding the common term of a power series is considered. A direct formula for evaluating this common term has been obtained. This formula provides useful expressions for direct evaluation of the number of partitions of a nonnegative integer and the partitions themselves. These expressions permit easily to work with power series, evaluate nth derivative of a composite function, calculate Bernoul­ li, Euler, Bell and other numbers, evaluate Bell polynomials, cycle index etc. Alternative expressions and some other new results for the truncated Bell polynomials have also been obtained. Some statistical applications of results under consideration and features of gen­ eralized probability generating functions are discussed.

The Cosmological Mass Distribution from Cayley Trees with Disorder

Abstract: We present a new approach to the statistics of the cosmic density field and to the mass distribution of high-contrast structures, based on the formalism of Cayley trees. Our approach includes in one random process both fluctuations and interactions of the density perturbations. We connect tree-related quantities, like the partition function or its generating function, to the mass distribution. The Press \& Schechter mass function and the Smoluchowski kinetic equation are naturally recovered as two limiting cases corresponding to independent Gaussian fluctuations, and to aggregation of high-contrast condensations, respectively. Numerical realizations of the complete random process on the tree yield an excess of large-mass objects relative to the Press \& Schechter function. When interactions are fully effective, a power-law distribution with logarithmic slope -2 is generated.

Fractional iteration of probability generating functions and imbedding discrete branching processes in continuous processes

Abstract: Fractional iteration of probability generating functions is investigated. In particular, conditions on the generating function of a Galton-Watson process that are necessary and sufficient for the process to admit imbedding in a continuous-time homogeneous Markov branching process are obtained. Necessary imbedding conditions formulated in terms of the several initial coefficients of the generating function are also obtained. The collection of all probability generating functions is partitioned, in accordance with a classification of branching processes, into subsets, and the latter are described as convex hulls of their extreme points. A description is given of the infinitesimal generators of distinguished semigroups of probability generating functions.Bibliography: 15 titles.

Waiting for HTHT.. .HT and geometric transforms

Abstract: It is shown how geometric transforms (probability generating functions) can be used to study the expected number of tosses until HTHT.. .HT.

An estimation method for the excess over the bocndaries in the sprt and its applications

Abstract: Since Wald developed the sequential probability ratio test, many studies have been done to approximate the characteristics of the test. One of the major efforts among them is to approximate the excess over the boundaries used in the test. In this paper the excess is approximated as a simple function of the parameter to be tested by using the condition of the test statistic immediately before the stopping time in normal and exponential cases. The use of the estimated excess shows good performances in estimating the operating characteristic function, the average sample number, and the probability mass function of the sarnple number. It also make it possible to determine the boundary values which can give the error probabilities close to the desired ones.

Self-Couplings and the concentration function

Abstract: The concentration function of a random variable may be estimated using couplings of the variable with itself.

Reducing third order cumulant spread in computer generated random numbers

Abstract: This paper describes a novel procedure for reducing the third order cumulant spread of random sequences with symmetric probability density function (PDF). The method is based on a new performance criterion related to the bispectral power of the random sequences. Computer simulations prove that the method can be used to generate a new set of 'whiter' random sequences.

Iterations of random and deterministic functions

Abstract: Letf be a probability generating function on [0, 1]. The convergence of its iteratesfn to fixed points is studied in this paper. Results include rates forf andf-1. Also iterates of independent identically distributed stable processes are studied and a trichotomy based on the order of the stability is established.

On the calculation of density functions for certain random geometric series

Abstract: Let be a regular stationary Markov pulse train A random geometric series based on {Xn } is given by the random variable where 0 < a < 1 This variable has a long history in the mathematical and engineering literature In applications it is of interest to be able to compute the probability density function of X In this paper a computational technique is derived to evaluate the probability density function of X It is shown that this method is numerically stable even in extreme conditions when other methods are impractical