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 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 discrete Ramos-Louzada distribution for asymmetric and over-dispersed data with leptokurtic-shaped: Properties and various estimation techniques with inference

2022 ◽  
Vol 7 (2) ◽  
pp. 1726-1741
Author(s):  
Ahmed Sedky Eldeeb ◽  
◽  
Muhammad Ahsan-ul-Haq ◽  
Mohamed. S. Eliwa ◽  
◽  
...  
Keyword(s):  
Count Data ◽  
Discrete Distribution ◽  
Real Data ◽  
Mass Function ◽  
Probability Mass Function ◽  
Data Sets ◽  
Estimation Techniques ◽  
Proposed Model ◽  
Probability Mass ◽  
Von Mises

<abstract> <p>In this paper, a flexible probability mass function is proposed for modeling count data, especially, asymmetric, and over-dispersed observations. Some of its distributional properties are investigated. It is found that all its statistical and reliability properties can be expressed in explicit forms which makes the proposed model useful in time series and regression analysis. Different estimation approaches including maximum likelihood, moments, least squares, Andersonӳ-Darling, Cramer von-Mises, and maximum product of spacing estimator, are derived to get the best estimator for the real data. The estimation performance of these estimation techniques is assessed via a comprehensive simulation study. The flexibility of the new discrete distribution is assessed using four distinctive real data sets ԣoronavirus-flood peaks-forest fire-Leukemia? Finally, the new probabilistic model can serve as an alternative distribution to other competitive distributions available in the literature for modeling count data.</p> </abstract>

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.

scholarly journals Discrete uniform and binomial distributions with infinite support

2020 ◽  
Vol 24 (23) ◽  
pp. 17517-17524 ◽  
Author(s):  
Andrey Pepelyshev ◽  
Anatoly Zhigljavsky
Keyword(s):  
Binomial Distribution ◽  
Numerical Study ◽  
Probability Distributions ◽  
Random Variable ◽  
Mass Function ◽  
Probability Mass Function ◽  
Binomial Distributions ◽  
Probability Mass ◽  
Discrete Uniform ◽  
Infinite Set

AbstractWe study properties of two probability distributions defined on the infinite set $$\{0,1,2, \ldots \}$$ { 0 , 1 , 2 , … } and generalizing the ordinary discrete uniform and binomial distributions. Both extensions use the grossone-model of infinity. The first of the two distributions we study is uniform and assigns masses $$1/\textcircled {1}$$ 1 / 1 to all points in the set $$ \{0,1,\ldots ,\textcircled {1}-1\}$$ { 0 , 1 , … , 1 - 1 } , where $$\textcircled {1}$$ 1 denotes the grossone. For this distribution, we study the problem of decomposing a random variable $$\xi $$ ξ with this distribution as a sum $$\xi {\mathop {=}\limits ^\mathrm{d}} \xi _1 + \cdots + \xi _m$$ ξ = d ξ 1 + ⋯ + ξ m , where $$\xi _1 , \ldots , \xi _m$$ ξ 1 , … , ξ m are independent non-degenerate random variables. Then, we develop an approximation for the probability mass function of the binomial distribution Bin$$(\textcircled {1},p)$$ ( 1 , p ) with $$p=c/\textcircled {1}^{\alpha }$$ p = c / 1 α with $$1/2<\alpha \le 1$$ 1 / 2 < α ≤ 1 . The accuracy of this approximation is assessed using a numerical study.

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.

Quantification of Low-Cycle Fatigue Life for a Gas Turbine Compressor Vane Carrier Under Varying Operating Conditions

2020 ◽  
Author(s):  
Zixi Han ◽  
Zixian Jiang ◽  
Sophie Ehrt ◽  
Mian Li
Keyword(s):  
Low Cycle Fatigue ◽  
Computational Cost ◽  
Mass Function ◽  
Operating Conditions ◽  
Probability Mass Function ◽  
Assessment Approach ◽  
Probability Mass ◽  
Gas Turbine Compressor ◽  
Operating Cycles ◽  
Operational Data

Abstract The design of a gas turbine compressor vane carrier (CVC) should meet mechanical integrity requirements on, among others, low-cycle fatigue (LCF). The number of cycles to the LCF failure is the result of cyclic mechanical and thermal strain effects caused by operating conditions on the components. The conventional LCF assessment is usually based on the assumption on standard operating cycles — supplemented by the consideration of predefined extreme operations and safety factors to compensate a potential underestimate on the LCF damage caused by multiple reasons such as non-standard operating cycles. However, real operating cycles can vary significantly from those standard ones considered in the conventional methods. The conventional prediction of LCF life can be very different from real cases, due to the included safety margins. This work presents a probabilistic method to estimate the distributions of the LCF life under varying operating conditions using operational fleet data. Finite element analysis (FEA) results indicate that the first ramp-up loading in each cycle and the turning time before hot-restart cycles are two predominant contributors to the LCF damage. A surrogate model of LCF damage has been built with regard to these two features to reduce the computational cost of FEA. Miner’s rule is applied to calculate the accumulated LCF damage on the component and then obtain the LCF life. The proposed LCF assessment approach has two special points. First, a new data processing technique inspired by the cumulative sum (CUSUM) control chart is proposed to identify the first ramp-up period of each cycle from noised operational data. Second, the probability mass function of the LCF life for a CVC is estimated using the sequential convolution of the single-cycle damage distribution obtained from operational data. The result from the proposed method shows that the mean value of the LCF life at a critical location of the CVC is significantly larger than the calculated result from the deterministic assessment, and the LCF lives for different gas turbines of the same class are also very different. Finally, to avoid high computational cost of sequential convolution, a quick approximation approach for the probability mass function of the LCF life is given. With the capability of dealing with varying operating conditions and noises in the operational data, the enhanced LCF assessment approach proposed in this work provides a probabilistic reference both for reliability analysis in CVC design, and for predictive maintenance in after-sales service.

ON SEVERAL PROPERTIES OF A CLASS OF PREFERENTIAL ATTACHMENT TREES—PLANE-ORIENTED RECURSIVE TREES

2020 ◽  
pp. 1-19
Author(s):  
Panpan Zhang
Keyword(s):  
Continuous Time ◽  
Preferential Attachment ◽  
Mass Function ◽  
Probability Mass Function ◽  
Zagreb Index ◽  
Exact Probability ◽  
Probability Mass ◽  
Proper Scaling ◽  
Recursive Trees ◽  
Degree Variable

In this paper, several properties of a class of trees presenting preferential attachment phenomenon—plane-oriented recursive trees (PORTs) are uncovered. Specifically, we investigate the degree profile of a PORT by determining the exact probability mass function of the degree of a node with a fixed label. We compute the expectation and the variance of degree variable via a Pólya urn approach. In addition, we study a topological index, Zagreb index, of this class of trees. We calculate the exact first two moments of the Zagreb index (of PORTs) by using recurrence methods. Lastly, we determine the limiting degree distribution in PORTs that grow in continuous time, where the embedding is done in a Poissonization framework. We show that it is exponential after proper scaling.

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