Showing papers on "Probability-generating function published in 1997"

Stationary and non-stationary probability density function for non-linear oscillators

Abstract: A method for the evaluation of the stationary and non-stationary probability density function of non-linear oscillators subjected to random input is presented. The method requires the approximation of the probability density function of the response in terms of C-type Gram-Charlier series expansion. By applying the weighted residual method, the Fokker-Planck equation is reduced to a system of non-linear first order ordinary differential equations, where the unknowns are the coefficients of the series expansion. Furthermore, the relationships between the A-type and C-type Gram-Charlier series coefficient are derived.

Start-Up Demonstration Tests Under Markov Dependence Model with Corrective Actions

Abstract: A general probability model for a start-up demonstration test is studied. The joint probability generating function of some random variables appearing in the Markov dependence model of the start-up demonstration test with corrective actions is derived by the method of probability generating function. By using the probability generating function, several characteristics relating to the distribution are obtained.

Probabilistic Scaling for the Numerical Inversion of Nonprobability Transforms

Abstract: It is known that probability density functions and probability mass functions usually can be calculated quite easily by numerically inverting their transforms (Laplace transforms and generating functions, respectively) with the Fourier-series method. Other more general functions can be substantially more difficult to invert, because the aliasing and roundoff errors tend to be more difficult to control. In this article we propose a simple new scaling procedure for nonprobability functions that is based on transforming the given function into a probability density function or a probability mass function and transforming the point of inversion to the mean. This new scaling is even useful for probability functions, because it enables us to compute very small values at large arguments with controlled relative error.

Probability Distribution Functions of Succession Quotas in the Case of Markov Dependent Trials

Abstract: The exact probability distribution functions (pdf's) of the sooner andlater waiting time random variables (rv's) for the succession quota problemare derived presently in the case of Markov dependent trials. This is doneby means of combinatorial arguments. The probability generating functions(pgf's) of these rv's are then obtained by means of enumerating generatingfunctions (enumerators). Obvious modifications of the proofs provideanalogous results for the occurrence of frequency quotas and such a resultis established regarding the pdf of a frequency and succession quotas rv.Longest success and failure runs are also considered and their jointcumulative distribution function (cdf) is obtained.

Estimating parameters for discrete distributions via the empirical probability generating function

Abstract: We consider parameter estimation for a family of discrete distributions characterized by probability generating functions (pgf's). Kemp and Kemp (1988) suggest estimators based on the empirical probability generating function (epgf) the methods involve solving estimating equations obtained by equating functionals of the epgf and pgf on a fixed, finite set of values. We derive asymptotic theory for these estimators and consider some examples. Graphical techniques based on the theory are shown to be useful for exploratory analysis

Eigenvalue expression for a batch markovian arrival process

Abstract: Consider a batch Markovian arrival process (BMAP) as the counting process of an underlying Markov process representing the state of environment. Such a process is useful for representing correlated inputs for example. They are used both as a modeling tool and as a theoretical device to represent and approximate superposition of input processes and complex large systems. Our objective is to consider the first and second moments of the counting process depending on time and state. Assuming that the probability generating functions of batch size are analytic, and that eigenvalues of the infinitesimal generator are simple, we derive an analytic diagonalization for the matrix generating function of the counting process. Our main result gives the time-dependent form of the first and second factorial moments of the counting process, which is represented by eigenvalues and eigenvectors of the matrix generating function of the batch size.

Family of Pearson discrete distributions generated by the univariate hypergeometric function 3F2(α1, α2, α3; γ1, γ2; λ)

Abstract: In this work we present a study of the Pearson discrete distributions generated by the hypergeometric function 3F2(α1, α2, α3;γ1, γ2; λ), a univariate extension of the Gaussian hypergeometric function, through a constructive methodology. We start from the polynomial coefficients of the difference equation that lead to such a function as a solution. Immediately after, we obtain the generating probability function and the differential equation that it satisfies, valid for any admissible values of the parameters. We also obtain the differential equations that satisfy the cumulants generating function, moments generating function and characteristic function, From this point on, we obtain a relation in recurrences between the moments about the origin, allowing us to create an equation system for estimating the parameters by the moment method. We also establish a classification of all possible distributions of such type and conclude with a summation theorem that allows us study some distributions belonging to this family. © 1997 by John Wiley & Sons, Ltd.

A wider class of lagrange distributions of the second kind

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.

On the recognition and structure of probability generating functions

Abstract: If $$ M\left( s \right) = 1 - {e^{ - \pi (s)}} $$ is a probability generating function, the coefficients π j in the MacLaurin expansion π(s) comprise a harmonic renewal sequence. A simple sufficient condition is given which ensures that a non-negative sequence is harmonic renewal. This condition covers the case of the limiting conditional law of a subcritical Markov branching process.

Can One Load a Set of Dice So That the Sum Is Uniformly Distributed

Abstract: Introduction Suppose we have two ordinary six-faced dice, and S is the sum of the numbers shown when the dice are rolled once. Then the random variable S can take any of the eleven integer values from 2 to 12. Can one load the dice so that S is uniformly distributed, i.e., Pr(S = i) = 1/11 for i = 2,3,...,12? The answer is no, as can be shown by an elegant probabilistic argument using only the simple inequality that x + 1/x ? 2 for all x > 0. (See Problem 52 in [4, p. 130].) Another (analytical) proof uses a polynomial factorization, as follows. Let pi and ri be the probabilities of the number i appearing on the first and second dice, respectively, for i = 1, 2, 3, 4, 5, 6. Suppose that the distribution of S is uniform. Then the probability generating function of S (for an exposition on probability generating functions, see [3, p. 177]) satisfies the following identity:

Support and plausibility degrees in generalized functional models

Abstract: By discussing several examples, the theory of generalized functional models is shown to be very natural for modeling some situations of reasoning under uncertainty. A generalized functional model is a pair (f, P) where f is a function describing the interactions between a parameter variable, an observation variable and a random source, and P is a probability distribution for the random source. Unlike traditional functional models, generalized functional models do not require that there is only one value of the parameter variable that is compatible with an observation and a realization of the random source. As a consequence, the results of the analysis of a generalized functional model are not expressed in terms of probability distributions but rather by support and plausibility functions. The analysis of a generalized functional model is very logical and is inspired from ideas already put forward by R.A. Fisher in his theory of fiducial probability.

Symbolic computation and exact distributions of nonparametric test statistics

Abstract: We show how to use computer algebra for computing exact distributions on nonparametric statistics. We give several examples of nonparametric statistics with explicit probability generating functions that can be handled this way. In particular, we give a new table of critical values of the Jonckheere-Terpstra test that extends tables known in the literature.