Probability-generating function

About: Probability-generating function is a research topic. Over the lifetime, 752 publications have been published within this topic receiving 9361 citations.

On fractional linear bounds for probability generating functions

Abstract: The best upper and lower bounds for any probability generating function with mean m and finite variance are derived within the family of fractional linear functions with mean m. These are often intractable and simpler bounds, more useful for practical purposes, are then constructed. Direct applications in branching and epidemic theories are briefly presented; a slight improvement of the bounds is obtained for infinitely divisible distributions. FRACTIONAL LINEAR FUNCTION; INFINITELY DIVISIBLE DISTRIBUTION; BOUNDS; BRANCHING PROCESS; S-I-S INFECTIOUS DISEASE

An approximate numerical solution for multiclass preemptive priority queues with general service time distributions

Abstract: In this paper an approximate numerical solution for a multiclass preemptive priority single server queue is developed The arrival process of each class follows a Poisson distribution The service time distribution must have a rational Laplace transform, but is otherwise arbitrary and may be different for different classes The work reported here was motivated by a desire to compute the equilibrium probability distribution of networks containing preemptive priority servers Such networks are frequently encountered when modeling computer systems, medical care delivery systems and communication networks We wish to use an iterative technique which constructs a series of two station networks consisting of one station from the original network and one “complementary” station whose behavior with respect to the original station mimics that of the rest of the network At each iteration, it is necessary to compute the equilibrium probability distribution of one or more preemptive priority queuesAlthough such queues have been studied for some time, the resulting solutions have most often been developed utilizing transforms or probability generating functions, eg Jaiswal [1968]; in many cases of interest, inversion has not been attempted Miller [1981] presented explicit solutions for two class priority queues but Miller's work, which is based on that of Neuts, is limited to exponential service times and two classes The approach presented here is applicable to many classes and to more general service time distributions than have previously been consideredThe algorithm utilizes a bootstrap approach, a concept borrowed from dynamic programming The solution for class 1 is trivial Once we have solved the system with k different classes, we have available all the necessary information to solve the system with k+l classes We shall assume that each class has a distinct service time distribution Gk, with mean gk and variance s2k Let class k have preemptive priority over class l if k 1, we consider a model with one machine whose service time distribution is Gk The breakdown rate of the machine is the sum of the arrival rates of all higher priority jobs The downtime or repairtime of the machine has mean gk-l and variance s2k-l; these parameters are the mean and variance of the busy period in a preemptive priority system with k-l classesThe first step in the solution of all the machine breakdown and repair models considered herein is to construct the infinitesimal generator

A Note on the Moment Generating Function

Abstract: A simple proof is given to show that there always exists a neighborhood of zero in which a moment generating function has a power series expansion. Thus, the relation between moments and derivatives of the moment generating function at zero can be obtained without resorting to postcalculus theorems.

Empirical examination of Edgeworth series

Abstract: The material presented is based on a numerical investigation that was made for five types of probability approximations which involve the first seven terms of the Edgeworth series expansion for the distribution of a continuous random variableT. For each approximation, the probability expressions considered in the investigation were Pr(T≦t), Pr(−t≦T≦t) and Pr(−t+1≦T≦t), whereT has zero mean, unit variance, and specified central momentsμ3,μ4,μ5. Computations were made for thoset values in the set −4.00(0.25) 4.00 that are pertinent for the probability expression being considered and for all combinations of the following values forμ3,μ4,μ5∶μ3=−2.0, −1.0, −0.5,0.0, 0.5, 1.0, 2.0;μ4=1, 2, 3, 5, 10;μ5=0.0, 3μ3−6.0, 3μ3, 3μ3+6.0. The principal results of this paper consist of a specification (for each approximation, probability expression, andμ3,μ4,μ5 combination) of limits ont such that within these limits the computed values of the probability expression are meaningful; that is, satisfy required monotonicity properties as a function oft and are neither negative nor greater than unity. Also the values of Pr(T≦0) and of Pr(−1.75≦T≦1.75) are listed for the cases considered. These results indicate that the types of approximations investigated are of doubtful usefulness for the situations examined; that is, for cases where the third and higher order moments of the random variable considered differ substantially from those for the normal variable having the same mean and variance.

Estimation of Mutation Rates from Fluctuation Experiments via Probability Generating Functions

Abstract: This paper calculates probability distributions modeling the Luria-Delbruck experiment. We show that by thinking purely in terms of generating functions, and using a 'backwards in time' paradigm, that formulas describing various situations can be easily obtained. This includes a generating function for Haldane's probability distribution due to Ycart. We apply our formulas to both simulated and real data created by looking at yeast cells acquiring an immunization to the antibiotic canavanine. This paper is somewhat incomplete, having been last significantly modified in March 29, 2014. However the first author feels that this paper has some worthwhile ideas, and so is going to make this paper publicly available.