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.

Methods of Analyzing Variability

Abstract: In an introductory part basic concepts from probability theory, and specifically from the theory of random processes, are introduced as a basis for the characterization of variability of hydrological time series, space processes and time-space processes. A partial characterization of the random process under study is adopted in accordance with three different schemes: Characterization by distribution function (one dimensional), Second moment characterization, and Karhunen–Loeve expansion, that is, a series representation in terms of random variables and deterministic functions of a random process. The article follows the same division into three major sections. In the first one, distribution functions of frequent use in hydrology are shortly described as well as the flow duration curve. The treatment of second-order moments includes covariance/correlation functions, spectral functions and semivariograms. They allow establishing the structure of the data in space and time and its scale of variability. They also give the possibility of testing basic hypothesis of homogeneity and stationarity. By means of normalization and standardization, data can be transformed into new data sets owing these properties. The section on Karhunen–Loeve expansion includes harmonic analysis, analysis by wavelets, principal component analysis, and empirical orthogonal functions. The characterization by series representation in its turn assumes homogeneity with respect to the variance–covariance function. It is as such a tool for analyzing spatial-temporal variability relative to the first- and second-order moments in terms of new sets of common orthogonal random functions. Keywords: random variable; process, vector; persistence; time series; integral scale; distribution function; correlation function; semivariogram; Karhunen–Loeve expansion

Performance evaluation of self-similar traffic in multimedia wireless communication networks with power saving class type III in IEEE 802.16e

Abstract: In this paper, we present a theoretical analysis to numerically evaluate the system performance of multimedia wireless communication networks with power saving class type III in IEEE 802.16e for self-similar traffic. Our model is based on system operations using a batch arrival, and we suppose the batch size to be a random variable following a Pareto(c, α) distribution in order to capture the self-similar property. By using a discrete-time embedded Markov chain, we derive the probability generating functions of the number of data frames and batches for when the busy period begins and for when the system is in a busy cycle. Using the first and higher moments of the probability generating functions, we give the averages and the standard deviation for the system performance in the diffusion approximation for the operation process of the system. In numerical results, we show the performance measures such as the energy saving ratio, plus the average and the standard deviation for the handover ratio with different system parameters as examples.

Random Number Generator Generating Random Numbers According to an Arbitrary Probability Density Function

Abstract: A method for generating a stream of random numbers which is representative of a probability distribution function, the method comprising receiving a set of K values x i (i=1, . . . K), thereby to define a range within which all of said values fall, and information indicative of the relative probability of each value under said probability distribution function, for each individual value x i (i=1, . . . K) generating a set of n i numbers uniformly distributed over a vicinity of said individual value x i , where n i is determined, using said information, to reflect the relative probability of said individual value x i , and where the vicinities of x i , for all i=1 . . . K partition said range within which all of said values fall, and providing a stream of numbers by randomly selecting numbers from a set S comprising the union of said sets of n i numbers, for i=1, . . . K.

A recursive method for computing the coefficients of the Gram-Charlier series

Abstract: The Gram-Charlier series is a known tool for approximating a probability density function when the moments or the cumulants of a random variable are known. A recursive procedure is presented which is well suited for the numerical computation of the coefficients of the series.