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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01 Jan 2016
TL;DR: In this paper, explicit expressions of probability functions and probability generating functions for mixed Poisson distributed discrete random variables are given corresponding to the following structure density functions:======generalized gamma, generalized shifted gamma and generalized shifted======beta.
Abstract: Explicit expressions of probability functions and probability generating
functions for mixed Poisson distributed discrete random variables
are given corresponding to the following structure density functions:
generalized gamma, generalized shifted gamma and generalized shifted
beta. A discrete symmetric distribution corresponding to a stochastic
process is approximated by a beta distribution in a more accurate manner.
A generalized Beta-Poisson distribution is obtained. The results
are useful in biological and economical problems. Special cases are also
mentioned. Graphs are drawn for probability functions showing the
modality for different values of the parameters. Transition intensities
can be easily obtained for the various cases discussed in this paper. Finally,
by utilizing the fact that probabilities sum to 1, we obtain some
new results for generalized hypergeometric functions.
TL;DR: In this paper , the behavior of a batch arrival single server retrial queueing model under three different vacation policies was studied, i.e., single vacation, multiple vacations, and at most J-vacations with impatient customers in general retrial times.
Abstract: This paper studies the behavior of a batch arrival single server retrial queueing model under three different vacation policies. Three types of vacation policies, single vacation, multiple vacations, and atmost J-vacations with impatient customers in general retrial times are considered. The probability generating function and marginal generating function of orbit size are obtained in a steady state. The stability condition for each vacation model is derived. Performance measures such as mean orbit size, mean system size, mean waiting time of a customer, and the probabilities of the server being in different states have also been determined. Based on performance characteristics, a comparative analysis is performed among the three vacations. Numerical illustrations are displayed to establish the consistency of the theory developed.
05 Jun 2013
TL;DR: In this article, the authors used the Graphical Evaluation and Review Technique (GERT) to obtain probability generating functions of the waiting time distributions of 1st, and th nonoverlapping and overlapping occurrences of the pattern, involving homogenous Markov dependent trials.
Abstract: We use the Graphical Evaluation and Review Technique (GERT) to obtain probability generating functions of the waiting time distributions of 1st, and th nonoverlapping and overlapping occurrences of the pattern , involving homogenous Markov dependent trials. GERT besides providing visual picture of the system helps to analyze the system in a less inductive manner. Mean and variance of the waiting times of the occurrence of the patterns have also been obtained. Some earlier results existing in literature have been shown to be particular cases of these results.
TL;DR: By using general series expansion method, this article used the more complicated base function {(t - t0)m e−nt| m = 0,1, 2, ; n = 1, 2} to study the differential equation V′(t) = 1 − V2(t), V(0) = 0.
Abstract: Abstract By using general series expansion method, this paper used the more complicated base function {(t - t0)m e−nt| m = 0,1, 2, ; n = 1, 2, } to study the differential equation V′(t) = 1 – V2(t), V(0) = 0. We could obtain a better result than the homotopy analysis method.
15 Dec 2012
TL;DR: In this article, it is shown that the fact that the cumulative probability over all values a random variable can assume has to be equal to one is not always feasible to check for without a profound knowledge of mathematics.
Abstract: Probability theory can be understood as a particular field in mathematics. Hence, it is only to be expected that it relies intensely on theory from analysis and algebra. For example, the fact that the cumulative probability over all values a random variable can assume has to be equal to one is not always feasible to check for without a profound knowledge of mathematics. Continuous probability distributions involve a good deal of analysis and the more sophisticated a distribution is, the more mathematics is necessary to handle it.
Keywords:
monotonic functions;
continuous function;
continuous;
derivative;
monotonically increasing;
strictly monotonic increasing;
A continuous function ;
monotonically decreasing;
strictly monotonic increasing;
integral;
integral;
factorial;
gamma function;
beta function;
Bessel function;
NIG;
characteristic function