Topic
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.
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TL;DR: In this article, the authors deduce a lower estimate for the distribution tail of a random variable in terms of its moment-generating function, without using complex analysis, and show that this lower estimate can be used to estimate the distribution of the random variable.
Abstract: Without using complex analysis, we deduce a lower estimate for the distribution tail of a random variable in terms of its moment-generating function.
8 citations
TL;DR: In this article, the problem of approximating an arbitrary probability generating function by a polynomial LN(S) -= Io rsN is considered, and it is shown that if the coefficients r are chosen so that L N( ) agrees with g( 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.
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
7 citations
TL;DR: In this paper, the class of modified power series distributions is widened by removing the restriction that the functions f(z) and g(z), as two probability generating functions defined on nonnegative integers, be probability generation functions.
Abstract: Starting with the second Lagrange expansion, with f(z) and g(z) as two probability generating functions defined on non-negative integers, Janardan and Rao( 1983) introduced a new class of discrete distributions called the Lagrange Distributions (LD2) of the second kind. In this note, this class of LD2 distributions is widened by removing the restriction that the functions f(z) and g(z) be probability generation functions. It is also shown that the class of modified power series distributions is a subclass of LD2.
7 citations
04 Jun 2012
TL;DR: The major aim of the paper is to estimate the impact of the interclass correlation in the arrival stream on the queueing performance of the system, in terms of the ( average) number of customers in the system and the (average) customer delay and customer waiting time.
Abstract: This paper considers a discrete-time queueing system with one server and two classes of customers. All arriving customers are accommodated in one queue, and are served in a First-Come-First-Served order, regardless of their classes. The total numbers of arrivals during consecutive time slots are i.i.d. random variables with arbitrary distribution. The classes of consecutively arriving customers, however, are correlated in a Markovian way, i.e., the probability that a customer belongs to a class depends on the class of the previously arrived customer. Service-time distributions are assumed to be general but class-dependent. We use probability generating functions to study the system analytically. The major aim of the paper is to estimate the impact of the interclass correlation in the arrival stream on the queueing performance of the system, in terms of the (average) number of customers in the system and the (average) customer delay and customer waiting time.
7 citations
TL;DR: In this paper, a random variable that can take on N discrete values is considered and the probability mass function is not assumed exactly known, rather the N probabilities have a prior N-dimensional Beta distribution.
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
7 citations