scholarly journals Statistical analysis of random walks on network

2021 ◽  
pp. 77-83
Author(s):  
A. Kalikova
Keyword(s):  
Random Walks ◽  
Mass Function ◽  
Hitting Time ◽  
Cumulative Distribution ◽  
Probability Mass Function ◽  
Cover Time ◽  
Arbitrary Graph ◽  
Commute Time ◽  
Starting Point ◽  
Analytical Formulas

This paper describes an investigation of analytical formulas for parameters in random walks. Random walks are used to model situations in which an object moves in a sequence of steps in randomly chosen directions. Given a graph and a starting point, we select a neighbor of it at random, and move to this neighbor; then we select a neighbor of this point at random, and move to it etc. It is a fundamental dynamic process that arise in many models in mathematics, physics, informatics and can be used to model random processes inherent to many important applications. Different aspects of the theory of random walks on graphs are surveyed. In particular, estimates on the important parameters of hitting time, commute time, cover time are discussed in various works. In some papers, authors have derived an analytical expression for the distribution of the cover time for a random walk over an arbitrary graph that was tested for small values of n. However, this work will show the simplified analytical expressions for distribution of hitting time, commute time, cover time for bigger values of n. Moreover, this work will present the probability mass function and the cumulative distribution function for hitting time, commute time.

scholarly journals Modified Recursions for a Class of Compound Distributions

1996 ◽  
Vol 26 (2) ◽  
pp. 213-224 ◽  
Author(s):  
Karl-Heinz Waldmann
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Mass Function ◽  
Cumulative Distribution ◽  
Probability Mass Function ◽  
Initial Value ◽  
Compound Distributions ◽  
Claim Frequency ◽  
Probability Mass ◽  
Stop Loss

AbstractRecursions are derived for a class of compound distributions having a claim frequency distribution of the well known (a,b)-type. The probability mass function on which the recursions are usually based is replaced by the distribution function in order to obtain increasing iterates. A monotone transformation is suggested to avoid an underflow in the initial stages of the iteration. The faster increase of the transformed iterates is diminished by use of a scaling function. Further, an adaptive weighting depending on the initial value and the increase of the iterates is derived. It enables us to manage an arbitrary large portfolio. Some numerical results are displayed demonstrating the efficiency of the different methods. The computation of the stop-loss premiums using these methods are indicated. Finally, related iteration schemes based on the cumulative distribution function are outlined.

scholarly journals A New Discrete Distribution Arising from a Generalised Random Game and Its Asymptotic Properties

2021 ◽  
pp. 11-20
Author(s):  
R. Frühwirth ◽  
R. Malina ◽  
W. Mitaroff
Keyword(s):  
Numerical Study ◽  
Asymptotic Properties ◽  
Discrete Distribution ◽  
Rayleigh Distribution ◽  
Mass Function ◽  
Cumulative Distribution ◽  
Probability Mass Function ◽  
Expected Gain ◽  
Probability Mass ◽  
Analytic Expression

The rules of a game of dice are extended to a ``hyper-die'' with \(n\in\mathbb{N}\) equally probable faces, numbered from 1 to \(n\). We derive recursive and explicit expressions for the probability mass function and the cumulative distribution function of the gain \(G_n\) for arbitrary values of \(n\). A numerical study suggests the conjecture that for \(n \to \infty\) the expectation of the scaled gain \(\mathbb{E}[{H_n}]=\mathbb{E} [{G_n/\sqrt{n}\,}]\) converges to \(\sqrt{\pi/\,2}\). The conjecture is proved by deriving an analytic expression of the expected gain \(\mathbb{E} [{G_n}]\). An analytic expression of the variance of the gain \(G_n\) is derived by a similar technique. Finally,  it is proved that \(H_n\) converges weakly to the Rayleigh distribution with scale parameter~1.

scholarly journals Improved Method for Approximating the Hazard Rate for Convolution Poisson Random Variables

2021 ◽  
Author(s):  
Andrei Volodin ◽  
ALYA AL MUTAIRI
Keyword(s):  
Hazard Rate ◽  
Cumulative Distribution Function ◽  
Random Variables ◽  
Saddlepoint Approximation ◽  
Mass Function ◽  
Rate Function ◽  
Cumulative Distribution ◽  
Probability Mass Function ◽  
Complicated Model ◽  
Probability Mass

In this study, we investigate the performance of the saddlepoint approximation of the probability mass function and the cumulative distribution function for the weighted sum of independent Poisson random variables. The goal is to approximate the hazard rate function for this complicated model. The better performance of this method is shown by numerical simulations and comparison with a performance of other approximation methods.

scholarly journals Regional Incomes Structure Analysis in Slovak Republic On the Basis of EU-SILC Data

2017 ◽  
Vol 64 (2) ◽  
pp. 171-185 ◽  
Author(s):  
Milan Terek
Keyword(s):  
Structure Analysis ◽  
Slovak Republic ◽  
Mass Function ◽  
Cumulative Distribution ◽  
Probability Mass Function ◽  
Sampling Weights ◽  
Empirical Cumulative Distribution Function ◽  
Empirical Probability ◽  
Probability Mass ◽  
Household Incomes

Abstract The paper deals with the regional incomes structure analysis in Slovak republic on the basis of European Union statistics on income and living conditions in Slovak republic data. The empirical probability mass function and empirical cumulative distribution function is constructed with aid of given sampling weights. On the basis of these functions the median, medial, standard deviation and population histogram of the whole gross household incomes for the whole Slovak republic and separately for eight Slovak regions are estimated and compared.

Fault Detection Probability Evaluation Approach in Combinational Circuits Using Test Set Generation Method

2018 ◽  
Author(s):  
Namita Arya ◽  
Amit Prakash Singh
Keyword(s):  
Fault Detection ◽  
Maximum Error ◽  
Mass Function ◽  
Cumulative Distribution ◽  
Probability Mass Function ◽  
Combinational Circuits ◽  
Occurrence Probability ◽  
Test Set ◽  
Probability Of Error ◽  
Multiple Faults

This paper introduces an approach that chooses the fault detection by calculating probabilities using probability mass function (pmf) and cumulative distribution function (CDF). This work used a method for multiple stuck-at faults by producing a new test pattern in combinational circuits. We assumed that existence of all multiple faults is only because of one single component that is faulty. A complete test set can be created by all possible single stuck-at faults in a combinational circuit using some combination of gates. The test set generation fault detection method is applied on two different 3-bit input variable and 4-bit input variable circuits. The probability of error occurrence is calculated at both 3-bit and 4-bit input variable circuits. The resulting feature is used to obtain maximum error occurrence probability to detect faults by the logic used that the complexity of the circuit is inversely proportional to the fault occurrence probability. Then again, undetectability is directly proportional to the complexity of the circuit. Therefore, finest feasible circuit should have large input variable components with less complexity to reduce the fault occurrence probability.

scholarly journals LOG-CONCAVITY OF COMPOUND DISTRIBUTIONS WITH APPLICATIONS IN OPERATIONAL AND ACTUARIAL MODELS

2019 ◽  
pp. 1-26
Author(s):  
F. G. Badía ◽  
C. Sangüesa ◽  
A. Federgruen
Keyword(s):  
Risk Model ◽  
Mass Function ◽  
Observation Time ◽  
Cumulative Distribution ◽  
Fundamental Result ◽  
Time Interval ◽  
Probability Mass Function ◽  
Cost Coefficient ◽  
Actuarial Risk ◽  
Algorithmic Approaches

AbstractWe establish that a random sum of independent and identically distributed (i.i.d.) random quantities has a log-concave cumulative distribution function (cdf) if (i) the random number of terms in the sum has a log-concave probability mass function (pmf) and (ii) the distribution of the i.i.d. terms has a non-increasing density function (when continuous) or a non-increasing pmf (when discrete). We illustrate the usefulness of this result using a standard actuarial risk model and a replacement model.We apply this fundamental result to establish that a compound renewal process observed during a random time interval has a log-concave cdf if the observation time interval and the inter-renewal time distribution have log-concave densities, while the compounding distribution has a decreasing density or pmf. We use this second result to establish the optimality of a so-called (s,S) policy for various inventory models with a stock-out cost coefficient of dimension [$/unit], significantly generalizing the conditions for the demand and leadtime processes, in conjunction with the cost structure in these models. We also identify the implications of our results for various algorithmic approaches to compute optimal policy parameters.

Customer Arrival Event Processing on Computer Simulation for Discrete Event System

2014 ◽  
Vol 513-517 ◽  
pp. 2133-2136
Author(s):  
Ming Hai Yao ◽  
Xiao Ji Chen ◽  
Lei Zuo
Keyword(s):  
Computer Simulation ◽  
Differential Equations ◽  
Effective Means ◽  
Discrete Event ◽  
Mass Function ◽  
Cumulative Distribution ◽  
Probability Mass Function ◽  
Simulation Techniques ◽  
Probability Mass ◽  
Ticket Machine

Discrete event systems are widely used in the production and life, it is difficult to use conventional differential equations, differential equations, and other models to describe, the theoretical analysis method is difficult to obtain analytical solutions, computer simulation techniques to solve these problems provides an effective means. Arrival event is a typical discrete system event; on arrival event handling is always one of the difficulties of computer simulation, in this paper, banking customer arrival system as an example to study. For banks queuing system, customers arrive to obey the parameter of Poisson distribution is, the probability mass function through the distribution curves and cumulative distribution function curves to study the distribution of customer arrival; construction of single-queue multi-server system of customer arrival event subroutine flow chart, and processing steps will be described. Content of this study, it is suitable for the developed area bank to adopt "number ticket machine" approach to service.

scholarly journals Detecting Different Road Infrastructural Elements Based on the Stochastic Characterization of Speed Patterns

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Mario Muñoz-Organero ◽  
Ramona Ruiz-Blázquez
Keyword(s):  
Total Variation ◽  
Traffic Congestion ◽  
Automatic Detection ◽  
Mass Function ◽  
Total Variation Distance ◽  
Statistical Moments ◽  
Probability Mass Function ◽  
Traffic Lights ◽  
Traffic Conditions ◽  
Variation Distance

The automatic detection of road related information using data from sensors while driving has many potential applications such as traffic congestion detection or automatic routable map generation. This paper focuses on the automatic detection of road elements based on GPS data from on-vehicle systems. A new algorithm is developed that uses the total variation distance instead of the statistical moments to improve the classification accuracy. The algorithm is validated for detecting traffic lights, roundabouts, and street-crossings in a real scenario and the obtained accuracy (0.75) improves the best results using previous approaches based on statistical moments based features (0.71). Each road element to be detected is characterized as a vector of speeds measured when a driver goes through it. We first eliminate the speed samples in congested traffic conditions which are not comparable with clear traffic conditions and would contaminate the dataset. Then, we calculate the probability mass function for the speed (in 1 m/s intervals) at each point. The total variation distance is then used to find the similarity among different points of interest (which can contain a similar road element or a different one). Finally, a k-NN approach is used for assigning a class to each unlabelled element.

HEAVY TRAFFIC APPROXIMATIONS FOR A SYSTEM OF INFINITE SERVERS WITH LOAD BALANCING

1999 ◽  
Vol 13 (3) ◽  
pp. 251-273 ◽  
Author(s):  
Philip J. Fleming ◽  
Burton Simon
Keyword(s):  
State Space ◽  
Heavy Traffic ◽  
Joint Probability ◽  
Mass Function ◽  
Accurate Estimate ◽  
The State ◽  
Probability Mass Function ◽  
Traffic Loads ◽  
Wide Range ◽  
Heavy Traffic Approximations

We consider an exponential queueing system with multiple stations, each of which has an infinite number of servers and a dedicated arrival stream of jobs. In addition, there is an arrival stream of jobs that choose a station based on the state of the system. In this paper we describe two heavy traffic approximations for the stationary joint probability mass function of the number of busy servers at each station. One of the approximations involves state-space collapse and is accurate for large traffic loads. The state-space in the second approximation does not collapse. It provides an accurate estimate of the stationary behavior of the system over a wide range of traffic loads.

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