Showing papers on "Probability-generating function published in 1993"

The Analysis and Optimization of Probability Functions

Abstract: A problem of probability function optimization is considered. This function represents probability that some random quantity depending on deterministic parameters does not exceed some given level. The problem is motivated by studies of safety domains and risk control problems in complex stochastic systems. For example, pollution control includes maximization of probability that some given levels of deposition at reception points are not exceeded. Optimization of probability function is performed over a given range of parameters. To solve the problem stochastic quasi-gradient method is applied under quasi-concavity assumption on functions and measures involved. Convergence and rate of convergence results are presented.

Empirical probability generating function

Abstract: A convenient approach to the statistical analysis of distributions for counts is possible using the empirical probability generating function. In this paper we give an overview of recent results and show the usefulness and advantages of this methodology. On one hand, there are some stochastic models in which the probability generating function arises naturally and therefore it seems reasonable to use its empirical counterpart. On the other hand, this statistical tool has demonstrated to be useful in the study of classical statistical problems of distributions for counts, especially in exploratory data analysis, rapid multi-parameter estimation and testing the goodness of fit. Our recommendation is to make allowance for the empirical probability generating function when dealing with statistical inference for discrete distributions.

Alternative numerical solutions of stationary queueing-time distributions in discrete-time queues: GI/G/1

Abstract: This paper gives closed-form expressions, in terms of the roots of certain equations, for the distribution of the waiting time in queue, Wq, in the steady-state for the discrete-time queue GI/G/1. Essentially, this is done by finding roots of the denominator of the probability generating function of Wq and then resolving the generating function into partial fractions. Numerical examples are given showing the use of the required roots, even when there is a large number of them. The method discussed in this paper avoids spectrum factorization and uses both closed- and non-closed forms of interarrival- and service-time distributions. Approximations for the tail probabilities in terms of one or three roots taken in ascending order of magnitude are also discussed. The exact computational results that can be obtained from the methods of this study should prove useful to both practitioners and queueing theorists dealing with bounds, inequalities, approximations, and simulation results.

On the soonest and latest waiting time distributions: succession quotas

Abstract: Let be a sequence of independent and identically distributed random variables with let ki. be a given positive integer. We define the random variable Ni to be the smallest integer so that a run in i’s of length ki has occurred in the subsequence . Then Ni’s are correlated variables. Also each N Ni has a geometric distribution of order k Ni (see Philippou et al. (1983)). The main objective of the present paper is to study the probability generating functions and hence the means and variances of the minimum and the maximum of these m correlated geometric distributions of different orders. It generalizes some of the earlier results in Ebneshahrashoob and Sobel (1990) and Ling (1990). As by products, some recurrence relations among characteristics of order statistics defined on an arbitrary set of random variables are also established.

First-passage probability of a random walk on a disordered one-dimensional lattice

Abstract: A method is presented which yields the exact solution of the generating function for the first-passage probability and related observables, for a one-dimensional random walk with random hopping rates on each site. A geometrical and statistical-mechanical interpretation of the problem is suggested and is used for obtaining the asymptotic properties of the occupation and first-passage probabilities. The class of values with the property of self-averaging is found.

Bayesian updating of an arbitrary discrete distribution under a special case of partial information

Abstract: In this paper we consider a random variable that can take on N discrete values. The probability mass function is not assumed exactly known, rather the N probabilities have a prior N-dimensional Beta distribution. Data of two types are obtained: i). actual values of the random variable, ii). observations where all that is known is that the random variable took on a value larger than a certain size. Formulae are developed for the posterior expected values of the N probabilities. Potential applications in the areas of preventive maintenance and inventory management are discussed and numerical illustrations are presented

Random pulse streams and their applications

Abstract: 1. Introduction to Probability. Definitions of probability. Conditional probability - total probability, Bayes' theorem. Binomial law - Bernoulli trials. Probability distribution function-random variable. Probability density function. Average characteristics of random variable: expectation and moments. Two random variables - joint and conditional distributions. Function of one and two random variables. Characteristic and moment generating functions. Law of large numbers. 2. Introduction to Random Signals and Spectral Analysis of Random Signals. Random signals and their fundamental characteristic. Fundamental linear transformations of random signals. Generalised spectrum analysis of random signals the transformation of Karhunen-Loeve. Harmonic analysis of stationary random signals. 3. The Main Characteristics of Point Streams. Description of point streams. Poisson streams. Regular streams. Stationary palm streams. Erlang streams. Attenuated streams. Alternative streams. 4. The Random Rectangular Pulse Stream and its Fundamental Characteristics. Description of random rectangular pulse streams. Classification of streams. Characteristics of conditional (truncated) streams. Segmental partition of pulses. Point partition of pulses. 5. Spectral and Correlation Analysis of Random Pulse Streams. The autocorrelation function of rectangular pulse stream. Power spectrum of a random signal. Power spectrum density of the rectangular pulse stream. Digital filtering of random pulse streams based on discrete signal transforms. 6. Coincidences of Random Pulse Streams. Methods of coincidence analysis. Average characteristics of the coincidence of n pulse streams. The probability density function of coincidence length. The limiting properties of the probability density function. Coincidence probability. Coincidence of exactly K streams from n streams - the K, n coincidence stream. The coincidence of at least K streams from n streams min (K, n) coincidence stream. The probability density function of a (K, n) coincidence stream. Some problems in deriving the probability density functions of aggregate streams. Decomposition of random pulse streams. Simulation of random pulse streams. 7. Some Applications of the Analysis of Random Pulse Streams. Characteristics of information transmission and the interval codes. Throughput of packets in random-access communications. Delay of packets in a random-access channel. Reliability characteristics of systems.

Annuity distributions: A new class of compound Poisson distributions

Abstract: A discrete random variable N is said to have an annuity distribution if its probabilities satisfy the recursion p n = p n -1 ( a + b/c n ), n = 1,2,3,… where a and b are real constants and c n = (1-e − n δ )/(e δ - 1), −∞ N satisfies a functional equation. This functional equation is used to prove that when N is an unbounded random variable then it is a compound Poisson random variable. It is also shown that the cumulants of N can be expressed as differences of Lambert series.

The effect of short production runs on csp-2

Abstract: This paper deals with the dynamic average outgoing quality AOQ(t) for CSP2. The main methods are transition probability flow graphs (TPFG), transition probability generating functions (TPGF) and probability generating functions (PGF).

On bounding Rician-type error probabilities and some applications

Abstract: The problem of efficiently evaluating the probability that the magnitude squared of one complex Gaussian random variable is less than the magnitude squared of another, possibly correlated, complex Gaussian random variable is addressed. This is known as an error probability of the Rician-type. In this paper, a bounding approach is taken. Tight upper and lower bounds are obtained, which together provide very accurate estimates of the actual error probability. Some applications of the bounds are presented to show their general applicability and to illustrate their tightness.

Generating function approach to quantum-to-classical correspondence I

Abstract: A method on the generating function, which produces time-evolution equations for moments of coordinate and momentum, is presented to study quantum-to-classical correspondence. A time-evolution equation for the quantal generating function is derived, which reduces to a classical one in the limit h → 0 . A quantal analogue of the classical distribution function is defined as the Fourier transform of the generating function. The quantal correction of the generating function is discussed. The relation between a stationary generating function and the energy eigenstates is discussed.