scholarly journals Non-uniform Estimates in the Approximation by the Irwin Law

2016 ◽  
Vol 55 (1) ◽  
pp. 112-118
Author(s):  
Kazimieras Padvelskis ◽  
Ruslan Prigodin
Keyword(s):  
Distribution Function ◽  
Remainder Term ◽  
Cumulative Distribution Function ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Type Condition ◽  
Uniform Estimates ◽  
Cumulative Distribution Functions ◽  
Cumulant Method ◽  
The Difference

We consider an approximation of a cumulative distribution function F(x) by the cumulative distributionfunction G(x) of the Irwin law. In this case, a function F(x) can be cumulative distribution functions of sums (products) ofindependent (dependent) random variables. Remainder term of the approximation is estimated by the cumulant method.The cumulant method is used by introducing special cumulants, satisfying the V. Statulevičius type condition. The mainresult is a nonuniform bound for the difference |F(x)-G(x)| in terms of special cumulants of the symmetric cumulativedistribution function F(x).

scholarly journals Distribution Function, Probability Generating Function and Archimedean Generator

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2108
Author(s):  
Weaam Alhadlaq ◽  
Abdulhamid Alzaid
Keyword(s):  
Distribution Function ◽  
Generating Function ◽  
Generating Functions ◽  
Probability Generating Function ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Archimedean Copulas ◽  
Cumulative Distribution Functions ◽  
Probability Generating Functions ◽  
Real Functions

Archimedean copulas form a very wide subclass of symmetric copulas. Most of the popular copulas are members of the Archimedean copulas. These copulas are obtained using real functions known as Archimedean generators. In this paper, we observe that under certain conditions the cumulative distribution functions on (0, 1) and probability generating functions can be used as Archimedean generators. It is shown that most of the well-known Archimedean copulas can be generated using such distributions. Further, we introduced new Archimedean copulas.

scholarly journals Asymptotics for the time of ruin in the war of attrition

2017 ◽  
Vol 49 (2) ◽  
pp. 388-410 ◽  
Author(s):  
Philip A. Ernst ◽  
Ilie Grigorescu
Keyword(s):  
Distribution Function ◽  
Explicit Formula ◽  
Cumulative Distribution Function ◽  
Cumulative Distribution ◽  
Fluid Limit ◽  
War Of Attrition ◽  
Time Of Ruin ◽  
Standard Normal ◽  
The Difference ◽  
The Time Of Ruin

AbstractWe consider two players, starting withmandnunits, respectively. In each round, the winner is decided with probability proportional to each player's fortune, and the opponent loses one unit. We prove an explicit formula for the probabilityp(m,n) that the first player wins. Whenm~Nx0,n~Ny0, we prove the fluid limit asN→ ∞. Whenx0=y0,z→p(N,N+z√N) converges to the standard normal cumulative distribution function and the difference in fortunes scales diffusively. The exact limit of the time of ruin τNis established as (T- τN) ~N-βW1/β, β = ¼,T=x0+y0. Modulo a constant,W~ χ21(z02/T2).

scholarly journals A Comparative Study on Computation of Cumulative Distribution Function in Predicting Time of Failure of Engineering Systems

2019 ◽  
Vol 11 (1) ◽  
Author(s):  
Gina Katherine Sierra Paez ◽  
Matthew Daigle ◽  
Kai Goebel
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Degradation Process ◽  
Cumulative Distribution ◽  
System State ◽  
Underlying Assumption ◽  
Hazard Zone ◽  
The Difference ◽  
Definition Of ◽  
Time Of Failure

Estimating accurate Time-of-Failure (ToF) of a system is key in making the decisions that impact operational safety and optimize cost. In this context, it is interesting to note that different approaches have been explored to tackle the problem of estimating ToF. The difference is in part characterized by different definitions of the hazard zones. The conventional definition for the cumulative distribution function (CDF) calculation is assumed to have well-defined hazard zones, that is, hazard zones defined as a function of the system state trajectory. An alternate method suggests the use of hazard zones defined as a function of the system state at time , instead of hazard zones defined as a function of system state up to and including time k (Acuña and Orchard 2018, 2017). This paper explores these differences and their impact on ToF estimation. Results for the conventional CDF definition indicated that, (i) the cumulative distribution function is always an increasing function of time, even when realizations of the degradation process are not monotonic, (ii) the sum of all probabilities is always 1 and does not need to be normalized, and (iii) all probabilities are positive and less than or equal to 1. Similar results are not observed for CDF calculation with hazard zones defined as a function only of the system state at time k. Results for ToF estimation using Acuña's definition differ, suggesting that there is an underlying assumption of independence in the hazard zone definition.  Therefore, we present an alternate definition of hazard zone which guarantees the properties of a well-defined CDF with a more straightforward ToF definition.

scholarly journals Computing the Stationary Distribution of Queueing Systems with Random Resource Requirements via Fast Fourier Transform

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 772 ◽  
Author(s):  
Valeriy A. Naumov ◽  
Yuliya V. Gaidamaka ◽  
Konstantin E. Samouylov
Keyword(s):  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Cumulative Distribution Function ◽  
Queueing Systems ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Queuing Systems ◽  
Cumulative Distribution Functions ◽  
Resource Requirements ◽  
Convolution Powers

Queueing systems with random resource requirements, in which an arriving customer, in addition to a server, demands a random amount of resources from a shared resource pool, have proved useful to analyze wireless communication networks. The stationary distributions of such queuing systems are expressed in terms of truncated convolution powers of the cumulative distribution function of the resource requirements. Discretization of the cumulative distribution function and the application of the fast Fourier transform are a traditional way of calculating convolutions. We suggest finding truncated convolution powers of the cumulative distribution functions by calculating the convolution powers of the truncated cumulative distribution functions via fast Fourier transform. This radically decreases computational complexity. We introduce the concept of resource load and investigate the accuracy of the proposed method at low and high resource loads. It is shown that the proposed method makes it possible to quickly and accurately calculate truncated convolution powers required for the analysis of queuing systems with random resource requirements.

DISSONANCE — A MEASURE OF VARIABILITY FOR ORDINAL RANDOM VARIABLES

2001 ◽  
Vol 09 (01) ◽  
pp. 39-53 ◽  
Author(s):  
RONALD R. YAGER
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Probability Distributions ◽  
Random Variables ◽  
Cumulative Distribution ◽  
General Properties

We look at the issue of obtaining a variance like measure associated with probability distributions over ordinal sets. We call these dissonance measures. We specify some general properties desired in these dissonance measures. The centrality of the cumulative distribution function in formulating the concept of dissonance is pointed out. We introduce some specific examples of measures of dissonance.

Estimating the cumulative distribution function for the linear combination of gamma random variables

2017 ◽  
Vol 20 (5) ◽  
pp. 939-951
Author(s):  
Amal Almarwani ◽  
Bashair Aljohani ◽  
Rasha Almutairi ◽  
Nada Albalawi ◽  
Alya O. Al Mutairi
Keyword(s):  
Distribution Function ◽  
Linear Combination ◽  
Cumulative Distribution Function ◽  
Random Variables ◽  
Cumulative Distribution
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