Probability-generating function

A probability-generating function is a power series representation of the probability mass function for discrete random variables that take integer values. It is denoted by G(z) and allows for the calculation of expected value and variance. The function is used to calculate moments for a probability discrete distribution.From: Some Structural Properties Related to the Borel-Taner Distribution and its’ Application [2023], Probability and Stochastic Modeling [2019], Applied Combinatorics [2019], An approach to non-homogenous phase-type distributions through multiple cut-points [2023]

Discrete Probability Distributions

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Published in Serkan Eryilmaz, Discrete Stochastic Models and Applications for Reliability Engineering and Statistical Quality Control, 2023

Serkan Eryilmaz

where ψ(z) is the probability generating function of discrete random variable Y.

Some Structural Properties Related to the Borel-Taner Distribution and its’ Application

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Published in American Journal of Mathematical and Management Sciences, 2023

Indranil Ghosh, Jordan Tanley

When deriving the structural properties of a distribution, it is necessary to identify the probability and moment generating functions to get some more information regarding the behavior of the underlying probability models. The probability generating function (p.g.f.) is a power series representation of the probability mass function that is effective for discrete random variables taking integer values. For this reason, the p.g.f. may be valuable in deriving some structural properties of the BT distribution. Let be the probability generating function of X. Then for values of s within the interval

An approach to non-homogenous phase-type distributions through multiple cut-points

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Published in Quality Engineering, 2023

Juan Eloy Ruiz-Castro, Christian Acal, Juan B. Roldán

To calculate the moments for a probability discrete distribution, the probability-generating function is built. This function is defined as where it converges absolutely at least for all complex numbers z with |z| ≤ 1; in many examples the radius of convergence is larger. From the derivation of this, the moments are obtained by evaluating in z = 1.

Probability-generating function

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Discrete Probability Distributions

Some Structural Properties Related to the Borel-Taner Distribution and its’ Application

An approach to non-homogenous phase-type distributions through multiple cut-points