Showing papers on "Probability-generating function published in 2011"

Sharp bounds by probability-generating functions and variable drift

Abstract: We introduce to the runtime analysis of evolutionary algorithms two powerful techniques: probability-generating functions and variable drift analysis. They are shown to provide a clean framework for proving sharp upper and lower bounds. As an application, we improve the results by Doerr et al. (GECCO~2010) in several respects. First, the upper bound on the expected running time of the most successful quasirandom evolutionary algorithm for the OneMax function is improved from 1.28n ln n to 0.982n ln n, which breaks the barrier of n ln n posed by coupon-collector processes. Compared to the classical 1+1-EA, whose runtime will for the first time be analyzed with respect to terms of lower order, this represents a speedup by more than a factor of e=2.71...

Dynamic Probability Power Flow of District Grid Containing Distributed Generation

Abstract: The power flow in the district grid varies with some randomicity especially distributed generations are integrated,such as wind power generation,photovoltaic generation etcAccording to the laws and the stochastic waving of the changes of wind energy,solar energy and electric power load,dynamic random variable was defined and its probability model was built by integrating the laws of the changes and random variableThe numeric characteristics of the dynamic random variable were calculated by simple calculation based on using semi-invariantAnd its probability density function was estimated through the approximate method on Gram-Charlier progressionSimulation tests were carried out on data of a district gridThe result indicates that the probability model of the dynamic random variable can comprise the laws and the stochastic wavingAnd the calculation method of dynamic probability power flow presented in this paper is feasible

Gambler’s ruin and winning a series by m games

Abstract: Two teams play a series of games until one team accumulates m more wins than the other. These series are fairly common in some sports provided that the competition has already extended beyond some number of games. We generalize these schemes to allow ties in the single games. Different approaches offer different advantages in calculating the winning probabilities and the distribution of the duration N, including difference equations, conditioning, explicit and implicit path counting, generating functions and a martingale-based derivation of the probability and moment generating functions of N. The main result of the paper is the determination of the exact distribution of N for a series of fair games without ties as a sum of independent geometrically distributed random variables and its approximation.

Kanter random variable and positive free stable distributions

Abstract: According to a representation due to M. Kanter, the density of some power of a positive stable distribution is a completely monotone function. In this paper, we first derive its representative Bernstein measure which also describes the law of some function of a uniform random variable, referred to below as the Kanter random variable. Then, the distribution function of the latter variable is written down and gives a more explicit description of the non commutative analogue of positive stable distributions in the setting of Voiculescu's free probability theory. Analytic evidences of the occurrence of the Kanter random variable in both the classical and the free settings conclude the exposition.

Bounds for the mean residual life function of a k -out-of- n system

Abstract: In the reliability studies, k-out-of-n systems play an important role. In this paper, we consider sharp bounds for the mean residual life function of a k-out-of-n system consisting of n identical components with independent lifetimes having a common distribution function F, measured in location and scale units of the residual life random variable Xt = (X−t|X > t). We characterize the probability distributions for which the bounds are attained. We also evaluate the so obtained bounds numerically for various choices of k and n.

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.

Analysis of a Two-Phase Queueing System with Impatient Customers and Multiple Vacations

Abstract: In this paper, we consider a two-phase queueing system with impatient customers and multiple vacations. Customers arrive at the system according to a Poisson process. They receive the first essential service as well as a second optional service. Arriving customers may balk with a certain probability and may depart after joining the queue without getting service due to impatience. Lack of service occurs when the server is on vacation or busy during the first phase of service. We analyze this model and derive the probability generating functions for the number of customers present in the system for various states of the server. We further obtain the closed-form expressions for various performance measures including the mean system sizes for various states of the server, the average rate of balking, the average rate of reneging, and the average rate of loss.

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.

Inverting generating functions with increased numerical precision — A computational experience

Abstract: In this paper, we consider the numerical inversion of a variety of generating functions (GFs) that arise in the area of engineering and non-engineering fields. Three classes of GFs are taken into account in a comprehensive manner: classes of probability generating functions (PGFs) that are given in rational and non-rational forms, and a class of GFs that are not PGFs. Among others, those PGFs that are not explicitly given but contain a number of unknowns are largely considered as they are often encountered in many interesting applied problems. For the numerical inversion of GFs, we use the methods of the discrete (fast) Fourier transform and the Taylor series expansion. Through these methods, we show that it is remarkably easy to obtain the desired sequence to any given accuracy, so long as enough numerical precision is used in computations. Since high precision is readily available in current software packages and programming languages, one can now lift, with little effort, the so-called Laplacian curtain that veils the sequence of interest. To demonstrate, we take a series of representative examples: the PGF of the number of customers in the discrete-time GeoX/Geo/c queue, the same in the continuous-time MX/D/c queue, and the GFs arising in the discrete-time renewal process.

On the relation between the distributions of stopping time and stopped sum with applications

Abstract: Let T be a stopping time associated with a sequence of independent random variables Z1;Z2;::: . By applying a suitable change in the probability measure we present relations between the moment or probability generating functions of the stopping time T and the stopped sum ST = Z1 +Z2 +::: +ZT . These relations imply that, when the distribution of ST is known, then the distribution of T is also known and vice versa. Applications are oered in order to illustrate the applicability of the main results, which also have independent interest. In the rst one we consider a random walk with exponentially distributed up and down steps and derive the distribution of its rst exit time from an interval ( a;b): In the second application we consider a series of samples from a manufacturing process and we let Zi;i 1, denoting the number of non-conforming products in the i-th sample. We derive the joint distribution of the random vector (T;ST ), where T is the waiting time until the sampling level of the inspection changes based on a k-run switching rule. Finally, we demonstrate how the joint distribution of (T;ST ) can be used for the estimation of the probability p of an item being defective, by employing an EM algorithm.

Apparatus for generating a probability graph model using a combination of variables and method for determining a combination of variables

Abstract: An apparatus and method for generating a probability graph model are provided. When generating a probability graph model using variable combinations, a variable combination that has a small amount of information may not generated, thereby reducing the amount of computation. The apparatus may acquire independent variables including a plurality of input variables corresponding to context information and an output variable corresponding to an inference result, and may determine a variable combination that is to be generated, based on the amount of information of each of variable combinations with respect to the output value, in which the variable combination is defined based on combining of the input variables.

Probability density formula of function of two-dimension random variable

Abstract: Aimed at simplifying the calculation of probability density of the two-dimension random variables function,this paper introduces the use of integral transform to the calculation formulae for probability density of two-dimension random variable function to provide new method designed for solving probability density of two-dimension random variable function.This method features a more simple operation and less difficult computationl than the distribution function method.

Reliability analysis method based on universal generating function

Abstract: Uncertainty is inevitable in practical engineering problems.Probabilistic reliability method is one of the uncertainty analysis methods.Integral calculation is needed in the probabilistic reliability method.However,it is very difficult or even impossible to obtain an analytical solution to the probabilistic reliability method based on sensitivity when involving high dimensionality and nonlinear integration.Therefore,a reliability analysis method based on the universal generating function is proposed.Firstly,the continuous random variables are discretized,and the probability mass function(PMF) is used to describe the discrete variables.Secondly,the operation algorithm by the universal generating function is used to calculate the PMF of the limit state function.Thirdly,the moments of the limit state function is analyzed by successively differentiating the PMF of the limit state function.Finally,an optimization design problem is formulated based on maximal entropy principle for estimating the probability density function and the cumulative distribution function of the limit state function.The sensitivity analysis and the calculation for solving the most probable failure points(MPP) are avoided in the proposed method,therefore,this method is proper for mixed-discrete and highly nonlinear problems.Comparison of the results from the proposed method and the Monte Carlo simulation method are presented to demonstrate the efficiency of the proposed method.

Shape control of conditional output probability density functions for linear stochastic systems with random parameters

Abstract: This paper presents a controller design for shaping conditional output probability density functions (pdf) for non-Gaussian dynamic stochastic systems whose coefficients are random and represented by their known pdfs The moment-generating function is applied to all the pdfs, leading to a simple mathematical relationship amongst all the transferred conditional pdfs of the system output and random parameters A new performance function is introduced and its minimisation is performed so as to design an optimal control input that makes the shape of the conditional output pdf follow a target distribution An example is included to illustrate the use of the algorithm

Runs and Patterns in a Sequence of Markov Dependent

Abstract: In this paper we consider a sequence of Markov dependent bivariate trials whose each component results in an outcome success (0) and failure (1) i.e. we have a sequence ,0 n X notes the num ber of occurrences of i-runs of length in the first component and 2 , i nk denotes the number of occurrences of i-runs of length k in the second component of Markov depen dent bivariate trials. Further we consider two patterns 1 i k i Y 2 i nn XY  number of occurrences of pattern denotes the   12   of length   12 k in the second) n components of bivariate trials. An algorithm is developed to evaluate the exact probability distributions of the vector random variables from their derived probability generating functions. Further some waiting time distribu- tions are studied using the joint distribution of runs. k first (

Some possible $q$-generalizations of harmonic numbers

Abstract: We study three different $q$-analogues of the harmonic numbers. As applications, we present some generating functions involving number theoretical functions and give the $q$-generalization of Gosper's exponential generating function of harmonic numbers. We involve also the $q$-gamma and $q$-digamma function.

The Gerber-Shiu Functions of a Risk Model with Two Classes of Claims and Random Income

Abstract: —In this paper, we consider a risk model involving two independent classes of insurance risks and random premium income. We assume that the premium income process is a Poisson Process, and the claim number processes are independent Poisson and generalized Erlang(n) processes, respectively. Both of the GerberShiu functions with zero initial surplus and the probability generating functions (p.g.f.) of the Gerber-Shiu functions are obtained.

Runs and Patterns in a Sequence of Markov Dependent Bivariate Trials

Abstract: In this paper we consider a sequence of Markov dependent bivariate trials whose each component results in an outcome success (0) and failure (1) i.e. we have a sequence {(Xn/Yn), n>=0} of S={(0/0),(0/1),(1/0),(1/1)}-valued Markov dependent bivariate trials. By using the method of conditional probability generating functions (pgfs), we derive the pgf of joint distribution of (X0n,k10,X1n,k11;Y0n,k20,Y1n,k21) where for i=0,1,Xin,k1i denotes the number of occurrences of i-runs of length k1i in the first component and Yin,k2i denotes the number of occurrences of i-runs of length k2i in the second component of Markov dependent bivariate trials. Further we consider two patterns Λ1 and Λ2 of lengths k1 and k2 respectively and obtain the pgf of joint distribution of (Xn,Λ 1,Yn,Λ2 ) using method of conditional probability generating functions where Xn,Λ1(Yn,Λ2) denotes the number of occurrences of pattern Λ1(Λ2 ) of length k1 (k2) in the first (second) n components of bivariate trials. An algorithm is developed to evaluate the exact probability distributions of the vector random variables from their derived probability generating functions. Further some waiting time distributions are studied using the joint distribution of runs.

An application of the stationary phase method for estimating probability densities of function derivatives

Abstract: We prove a novel result wherein the density function of the gradients---corresponding to density function of the derivatives in one dimension---of a thrice differentiable function S (obtained via a random variable transformation of a uniformly distributed random variable) defined on a closed, bounded interval \Omega \subset R is accurately approximated by the normalized power spectrum of \phi=exp(iS/\tau) as the free parameter \tau-->0. The result is shown using the well known stationary phase approximation and standard integration techniques and requires proper ordering of limits. Experimental results provide anecdotal visual evidence corroborating the result.

Moment inequalities for hybrid events with fuzziness and randomness

Abstract: In many cases, fuzziness and randomness simultaneously appear in a system. Hybrid variable is a tool to describe this phenomena. Fuzzy random variable and random fuzzy variable are instances of hybrid variable. In order further to discuss the properties of hybrid variables, several useful moment inequalities for hybrid variables are established based on the concepts of chance measures in this paper. This may enlarge the applications of hybrid variable and chance theory.

Reliability analysis based on combination of universal generating function and discrete approach

Abstract: Uncertainty exists in the engineering practices widely. Since a multidimensional integration problem should be dealt with during the process of reliability-based analysis and design, it is the key problem to develop new method to improve the efficiency and accuracy for the reliability-based analysis and design in the complex systems. Hence a new method is proposed, and the procedure of the proposed method is summarized as follows. First, transform continuous random variable into discrete random variables modeled by probability mass function (PMF). The PMF of a limit-state function can be acquired through universal generating function (UGF) and different moments can be calculated by using derivative. Second, maximum entropy principle is used to calculate the probability density function (PDF) of the limit-state function. The proposed method, based on the PMF and UGF, is suitable for the cases that discrete variables exist in the system and the limit-state function is a highly non-linear problem. The reason is that the proposed method needs neither derivative nor the most probable point (MPP) search. A numerical example is provided to demonstrate the effectiveness of the proposed method, and furthermore a comparison is made between the results from the proposed method and Monte Carlo simulation (MCS).

Non-convergence in probability of the overlapping function.

Abstract: Let us consider the set of n-length sequences on a countable alphabet. We consider the function that gives the maximum size of an overlap that wn has with itself, and denote it by Sn(wn), and call it the overlapping function. For n-sized IID sequences in our alphabet, we showed that the overlapping function converges in distribution, when n goes to infinity AbLa [1]. In this work, we show that, on the same conditions, the convergence of Sn cannot be in probability. We also show some behavior of the expectation of Sn and its limit as functions of the parameter space. Moreover, we present some bounds for E(Sn) and its limit.