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.

Trinomial random walk with one or two imperfect absorbing barriers

Abstract: Trinomial random walk, with one or two barriers, on the non-negative integers is discussed. At the barriers, the particle is either annihilated or reflects back to the system with respective probabilities 1 − ρ, ρ at the origin and 1 − ω, ω at L, 0 ≤ ρ,ω ≤ 1. Theoretical formulae are given for the probability distribution, its generating function as well as the mean of the time taken before absorption. In the one-boundary case, very qualitatively different asymptotic forms for the result, depending on the relationship between transition probabilities and the annihilation probability, are obtained.

A Characterization of the Members of a Subfamily of Power Series Distributions

Abstract: This paper discusses a characterization of the members of a subfamily of power series distributions when their probability generating functions satisfy the functional equation where a, b and c are constants and is the derivative of f.

Distribution and double generating function of number of patterns in a sequence of Markov dependent multistate trials

Abstract: In this manuscript, the dual relationship between the probability of number of runs and patterns and the probability of waiting time of runs and patterns in a sequence of multistate trials has been studied via double generating functions and recursive equations. The results, which are established under different assumptions on patterns, underlying sequences and counting schemes, are extensions of Koutras’s results (1997, Advances in Combinatorial Methods and Applications to Probability and Statistics, Boston: Birkhauser). As byproducts, the exact distributions of the longest-run statistics are also derived. Numerical examples are provided for illustrating the theoretical results.

Generalized Moment Generating Functions of Random Variables and Their Probability Density Functions

Abstract: This paper seeks to develop a generalized method of generating the moments of random variables and their probability distributions. The Generalized Moment Generating Function is developed from the existing theory of moment generating function as the expected value of powers of the exponential constant. The methods were illustrated with the Beta and Gamma Family of Distributions and the Normal Distribution. The methods were found to be able to generate moments of powers of random variables enabling the generation of moments of not only integer powers but also real positive and negative powers. Unlike the traditional moment generating function, the generalized moment generating function has the ability to generate central moments and always exists for all continuous distribution but has not been developed for any discrete distribution.

PDF estimation via characteristic function and an orthonormal basis set

Abstract: The efficacy of estimating the probability density function through, first, reconstruction of the characteristic function and, second, through use of a series approximation based on a Hermite orthonormal basis set, is shown. Both approaches yield a lower integrated error than a benchmark uniform kernel density function for the case of a probability density function with a significant tail. For a Hermite series it is shown that a scaling parameter can be set according to the root mean square value of the data to obtain optimum error performance. The Hermite series estimator is applied to approximating the probability density function evolution of a generalized shot noise process and to estimation of a bimodal probability density function.