Showing papers on "Probability-generating function published in 2007"

Application of Tauberian Theorem to the Exponential Decay of the Tail Probability of a Random Variable

Abstract: In this correspondence, we give a sufficient condition for the exponential decay of the tail probability of a nonnegative random variable. We consider the Laplace-Stieltjes transform of the probability distribution function of the random variable. We present a theorem, according to which if the abscissa of convergence of the LS transform is negative finite and the real point on the axis of convergence is a pole of the LS transform, then the tail probability decays exponentially. For the proof of the theorem, we extend and apply so-called a finite form of Ikehara's complex Tauberian theorem by Graham-Vaaler.

A discrete-time priority queue with train arrivals

Abstract: We analyze a discrete-time priority queue with train arrivals. Messages of a variable number of fixed-length packets belonging to two classes arrive to the queue at the rate of one packet per slot. We assume geometrically distributed message lengths. Packets of the first class have transmission priority over the packets of the other class. By using probability generating functions, some performance measures such as the moments of the packet delay are calculated. The impact of the priority scheduling discipline and the correlation in the arrival process is shown by some numerical examples.

Dynamics of epidemics on random networks.

Abstract: This paper examines how diseases on random networks spread in time. The disease is described by a probability distribution function for the number of infected and recovered individuals, and the probability distribution is described by a generating function. The time development of the disease is obtained by iterating the generating function. In cases where the disease can expand to an epidemic, the probability distribution function is the sum of two parts; one that is static at long times, and another whose mean grows exponentially. The time development of the mean number of infected individuals is obtained analytically. When epidemics occur, the probability distributions are very broad, and the uncertainty in the number of infected individuals at any given time is typically larger than the mean number of infected individuals.

Priority queueing systems: from probability generating functions to tail probabilities

Abstract: Obtaining (tail) probabilities from a transform function is an important topic in queueing theory. To obtain these probabilities in discrete-time queueing systems, we have to invert probability generating functions, since most important distributions in discrete-time queueing systems can be determined in the form of probability generating functions. In this paper, we calculate the tail probabilities of two particular random variables in discrete-time priority queueing systems, by means of the dominant singularity approximation. We show that obtaining these tail probabilities can be a complex task, and that the obtained tail probabilities are not necessarily exponential (as in most `traditional' queueing systems). Further, we show the impact and significance of the various system parameters on the type of tail behavior. Finally, we compare our approximation results with simulations.

Path-probability density functions for semi-Markovian random walks.

Abstract: In random walks, the path representation of the Green's function is an infinite sum over the length of path probability density functions (PDFs). Recently, a closed-form expression for the Green's function of an arbitrarily inhomogeneous semi-Markovian random walk in a one-dimensional (1D) chain of L states was obtained by utilizing path-PDFs calculations. Here we derive and solve, in Laplace space, the recursion relation for the n order path PDF for the same system. The recursion relation relates the n order path PDF to L/2 (round towards zero for an odd L) shorter path PDFs and has n independent coefficients that obey a universal formula. The z transform of the recursion relation straightforwardly gives the generating function for path PDFs, from which we recover the Green's function of the random walk, but, moreover, derive an explicit expression for any path PDF of the random walk. These expressions give the most detailed description of arbitrarily inhomogeneous semi-Markovian random walks in 1D.

Probability density function separator, probability density function separating method, program, tester, bit error rate measurement device, electronic device, and jitter transfer function measurement device

Abstract: A probability density function separator for separating a predetermined component from a given probability density function comprises: a domain converting section for converting the given probability density function into the spectrum of a frequency domain; and a fixed component calculating section for multiplying a first null frequency of the spectrum of the frequency domain by a multiplier coefficient corresponding to the type of the distribution of a fixed component included in the given probability density function so as to calculate the peak-to-peak value of the probability density function of the fixed component

Moment complete convergence for B-valued random elements

Abstract: In this paper,the slowly varying function and a class of functions indicated with S are introducedAnd then obtained under some conditions of probability are some sufficient conditions for moment complete convergence for independent non-identically distributed random elements in a separable Banach space which are stochastically bounded by a positive random variable

On the Waiting Time for the First Success Run

Abstract: Let k and m are positive integers with k ≥ m. The probability generating function of the waiting time for the first occurrence of consecutive k successes in a sequence of m-th order Markov dependent trials is given as a function of the conditional probability generating functions of the waiting time for the first occurrence of consecutive m successes. This provides an efficient algorithm for obtaining the probability generating function when k is large. In particular, in the case of independent trials a simple relationship between the geometric distribution of order k and the geometric distribution of order k−1 is obtained.

Coverage Probability of Random Intervals

Abstract: In this paper, we develop a general theory on the coverage probability of random intervals defined in terms of discrete random variables with continuous parameter spaces. The theory shows that the minimum coverage probabilities of random intervals with respect to corresponding parameters are achieved at discrete finite sets and that the coverage probabilities are continuous and unimodal when parameters are varying in between interval endpoints. The theory applies to common important discrete random variables including binomial variable, Poisson variable, negative binomial variable and hypergeometrical random variable. The theory can be used to make relevant statistical inference more rigorous and less conservative.

Probability density function isolating device, probability density function isolating method, test device, bit error ratio measuring device, electronic device, and program

Abstract: Provided is a probability density function isolating device for isolating a predetermined component from a given probability density function. The device includes: a region conversion unit for receiving a probability density function and converting the probability density function into a frequency region spectrum; and a definite component calculation unit for multiplying a first null frequency of the frequency region spectrum by a multiplication coefficient based on a distribution type of a definite component contained in the given probability density function so as to calculate a peak-to-peak value of the probability density function of the definite component.

Probabilistic Transformation Method in Reliability Analysis

Abstract: A technique is presented in order to evaluate the probability of failure in analytical form instead of approximation methods like FORM/SORM and no series expansion is involved in this expression This technique is based on the Finite Element Method to obtain the expression of the response of stochastic systems, and the transformation of random variables to obtain the probability density function of the response The transformation technique evaluates the probability density function (pdf) of the system output by multiplying the input pdf by the Jacobian of the inverse mechanical function This approach has the advantage of giving directly the whole density function of the response in closed form, which is very helpful for reliability analysis

Geometrical Domain of Spin 1/2 Probability Mass Function

Abstract: The quantum analogue of the classical characteristic function for a spin 1/2 assembly is considered and the probability mass function of the random vector associated with the assembly is derived. It is seen that the positive regions of Wigner and Margenau-Hill quasi distributions for the three components of spin, correspond to a trivariate probability mass function. We identify the domain of these positive regions as an Octahedron inscribed in the Bloch sphere with its vertices on the surface of the sphere. It is in this domain that a quantum characteristic function characterizing the quasi distribution, admits a probability mass function in IR^3 . It is also observed that the classical variates X1, X2, X3 corresponding to the 3 spin operators \sigma_1,\sigma_2,\sigma_3 in the domain, are independent iff the Bloch vector lies on any one of the axes.

Estimate of the concentration function for a class of additive functions

Abstract: We show how methods for estimating the concentration function of the sum of independent random variables can be used to obtain estimates of the concentration function for the values of additive functions under suitable conditions. Previously obtained estimates of the concentration function are consequences of the estimate obtained in the present paper for functions from of the class under consideration.

Applying the Moment Generating Functions to the Study of Probability Distributions

Abstract: In this paper, we describe a tool to aid in proving theorems about random variables, called the moment generating function, which converts problems about probabilities and expectations into problems from calculus about function values and derivates. We show how the moment generating function determinates the moments and how the moments can be used to recover the moment generating function. Using of moment generating functions to find distributions of functions of random variables is presented. A standard form of the central limit theorem is also stated and proved.

Random and exhaustive generation of permutations and cycles

Abstract: In 1986 S. Sattolo introduced a simple algorithm for uniform random generation of cyclic permutations on a fixed number of symbols. This algorithm is very similar to the standard method for generating a random permutation, but is less well known. We consider both methods in a unified way, and discuss their relation with exhaustive generation methods. We analyse several random variables associated with the algorithms and find their grand probability generating functions, which gives easy access to moments and limit laws.

On the gap probability generating function at the spectrum edge in the case of orthogonal symmetry

Abstract: The gap probability generating function has as its coefficients the probability of an interval containing exactly $k$ eigenvalues. For scaled random matrices with orthogonal symmetry, and the interval at the hard or soft spectrum edge, the gap probability generating functions have the special property that they can be evaluated in terms of Painleve transcendents. The derivation of these results makes use of formulas for the same generating function in certain scaled, superimposed ensembles expressed in terms of its correlation functions. It is shown that by a judicious choice of the superimposed ensembles, the scaled limit necessary to derive these formulas can be rigorously justified by a straight forward analysis.

Probability generating function of GPED2

Abstract: The Probability generating function of a random variable which has Generalized Polya Eggenberger Distribution of the second kind (GPED 2) is obtained. The probability density function of the range R, in random sampling from a uniform distribution on (k, l) and exponential distribution with parameter λ is obtained, when the sample size is a random variable from GPED 2. The results of Bazargan-Lari (2004) follow as special cases.

Asymptotics of the probability of values of random Boolean expressions

Abstract: The random Boolean expressions are considered that are obtained by the random and independent substitution with the probabilities p and 1 − p of the constantly one function and constantly zero function for variables of repetition-free formulas over a given basis. The probability is studied that the expressions are equal to one. It is shown that, for each finite basis and p ∊ (0, 1), this probability tends to some finite limit P 1(p) as the length of an expression grows. Explicit representation of the probability function P 1(p) is found for all finite bases, the analytic properties of this function are studied, and its behavior is investigated in dependence on the properties of the basis.

Large Deviations: Advanced Probability for Undergrads.

Abstract: In the branch of probability called "large deviations," rates of convergence (e.g. of the sample mean) are considered. The theory makes use of the moment generating function. So, particularly for sums of independent and identically distributed random variables, the theory can be made accessible to senior undergraduates after a first course in stochastic processes. This paper describes a directed independent study in large deviations offered to a strong senior, providing a sample outline and discussion of resources. Learning points are also highlighted.

Moments of random allocation processes reaching a boundary

Abstract: In this paper we develop some results presented by Gani (2004), deriving moments for random allocation processes These moments correspond to the allocation processes reaching some domain boundary Exact formulae for means, variances, and probability generating functions as well as some asymptotic formulae for moments of random allocation processes are obtained A special choice of the asymptotics and of the domain allows us to reduce a complicated numerical procedure to a simple asymptotic one

A note on three stochastic processes with immigration

Abstract: Three stochastic processes, the birth, death and birth-death processes, subject to immigration can be decomposed into the sum of each process in the absence of immigration and an independent process. We examine these independent processes through their probability generating functions (pgfs) and derive their expectations.

Stochastic random variables generating method for use in e.g. power line communications of motor vehicle, involves generating random variables by quantile transformation of distribution function of stored measuring data

Abstract: The method involves receiving measuring data of dimensions that are described by properties and/or characteristics of interfering impulse during operation of a motor vehicle. The received data and corresponding frequency distribution are stored in a distribution table. Random variables for the dimension with the distribution corresponding to a density function of the received data are generated based on the stored data. The variables are generated by a quantile transformation of a distribution function of the data. A content of the table is optimized from the generation of the variables. An independent claim is also included for a device for generating stochastic random variables for the emulation of interfering impulses on a motor vehicle electrical system.