scholarly journals On Some Compound Random Variables Motivated by Bulk Queues

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Romeo Meštrović
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Mass Function ◽  
Probability Mass Function ◽  
Time Of Arrival ◽  
Uniform Random Variable ◽  
Wide Range ◽  
Bulk Queue ◽  
Probability Mass ◽  
Number Of Customers

We consider the distribution of the number of customers that arrive in an arbitrary bulk arrival queue system. Under certain conditions on the distributions of the time of arrival of an arriving group (Y(t)) and its size (X) with respect to the considered bulk queue, we derive a general expression for the probability mass function of the random variableQ(t)which expresses the number of customers that arrive in this bulk queue during any considered periodt. Notice thatQ(t)can be considered as a well-known compound random variable. Using this expression, without the use of generating function, we establish the expressions for probability mass function for some compound distributionsQ(t)concerning certain pairs(Y(t),X)of discrete random variables which play an important role in application of batch arrival queues which have a wide range of applications in different forms of transportation. In particular, we consider the cases whenY(t)and/orXare some of the following distributions: Poisson, shifted-Poisson, geometrical, or uniform random variable.

scholarly journals COMPOUND RANDOM VARIABLES

2004 ◽  
Vol 18 (4) ◽  
pp. 473-484 ◽  
Author(s):  
Erol Peköz ◽  
Sheldon M. Ross
Keyword(s):  
Arbitrary Function ◽  
Negative Binomial ◽  
Random Variables ◽  
Random Variable ◽  
Mass Function ◽  
Probability Mass Function ◽  
Probabilistic Proof ◽  
Recursive Formulas ◽  
Probability Mass ◽  
Positive Compound

We give a probabilistic proof of an identity concerning the expectation of an arbitrary function of a compound random variable and then use this identity to obtain recursive formulas for the probability mass function of compound random variables when the compounding distribution is Poisson, binomial, negative binomial random, hypergeometric, logarithmic, or negative hypergeometric. We then show how to use simulation to efficiently estimate both the probability that a positive compound random variable is greater than a specified constant and the expected amount by which it exceeds that constant.

scholarly journals λ-Analogues of Stirling polynomials of the first kind and their applications

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Han Young Kim ◽  
Sung-Soo Pyo
Keyword(s):  
Recurrence Relations ◽  
Random Variable ◽  
Stirling Numbers ◽  
Mass Function ◽  
Probability Mass Function ◽  
Discrete Random Variable ◽  
Special Polynomials ◽  
Binomial Distributions ◽  
Natural Extensions ◽  
Probability Mass

AbstractRecently, λ-analogues of Stirling numbers of the first kind were studied. In this paper, we introduce, as natural extensions of these numbers, λ-Stirling polynomials of the first kind and r-truncated λ-Stirling polynomials of the first kind. We give recurrence relations, explicit expressions, some identities, and connections with other special polynomials for those polynomials. Further, as applications, we show that both of them appear in an expression of the probability mass function of a suitable discrete random variable, constructed from λ-logarithmic and negative λ-binomial distributions.

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.

PRESERVATION OF LOG-CONCAVITY UNDER CONVOLUTION

2017 ◽  
Vol 32 (4) ◽  
pp. 567-579
Author(s):  
Tiantian Mao ◽  
Wanwan Xia ◽  
Taizhong Hu
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Necessary Conditions ◽  
Random Variables ◽  
Mass Function ◽  
Probability Mass Function ◽  
Sufficient And Necessary Conditions ◽  
Probability Mass ◽  
Family Of Distributions

Log-concave random variables and their various properties play an increasingly important role in probability, statistics, and other fields. For a distribution F, denote by 𝒟F the set of distributions G such that the convolution of F and G has a log-concave probability mass function or probability density function. In this paper, we investigate sufficient and necessary conditions under which 𝒟F ⊆ 𝒟G, where F and G belong to a parametric family of distributions. Both discrete and continuous settings are considered.

AN ASYMPTOTIC EXPANSION FOR THE INSPECTION PARADOX

2005 ◽  
Vol 20 (1) ◽  
pp. 87-94
Author(s):  
J. E. Angus
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Random Variables ◽  
Mass Function ◽  
Probability Mass Function ◽  
Number Of Children ◽  
The Family ◽  
Probability Mass ◽  
Families With Children ◽  
Independent And Identically Distributed

Suppose that there arenfamilies with children attending a certain school and that the number of children in these families are independent and identically distributed random variables, each with probability mass functionP{X=j} =pj,j≥ 1, with finite mean μ = [sum ]j≥1jpj. If a child is selected at random from the school andXIis the number of children in the family to which the child belongs, it is known that limn→∞P{XI=j} =jpj/μ,j≥ 1. Here, asymptotic expansions forP{XI=j} are developed under the conditionE|X|3< ∞.

Evaluating Best-Case and Worst-Case Coefficients of Variation when Bounds Are Available

1992 ◽  
Vol 6 (3) ◽  
pp. 309-322 ◽  
Author(s):  
George S. Fishman ◽  
David S. Rubin
Keyword(s):  
Network Reliability ◽  
Random Variable ◽  
Mass Function ◽  
Probability Mass Function ◽  
Lower And Upper Bounds ◽  
Worst Case ◽  
The Monte Carlo Method ◽  
Coefficients Of Variation ◽  
Probability Mass ◽  
Bounded Random Variable

This paper describes a procedure for computing tightest possible best-case and worst-case bounds on the coefficient of variation of a discrete, bounded random variable when lower and upper bounds are available for its unknown probability mass function. An example from the application of the Monte Carlo method to the estimation of network reliability illustrates the procedure and, in particular, reveals considerable tightening in the worst-case bound when compared to the trivial worst-case bound based exclusively on range.

Weibull Distributed Stress-Dependent Strength Analysis of Aeroengine Alloy Using Lagrange Factor Polynomial

2010 ◽  
Author(s):  
Chun Nam Wong ◽  
Hong-Zhong Huang ◽  
Jingqi Xiong ◽  
Tianyou Hu
Keyword(s):  
Random Variable ◽  
Mass Function ◽  
Strength Distribution ◽  
Interference Model ◽  
Strength Analysis ◽  
Probability Mass Function ◽  
Application Range ◽  
Probability Mass ◽  
Stress Dependent ◽  
Mean Function

In this paper, the unilateral dependency of strength on stress is taken into account. And the stress-dependent strength is represented by a discrete random variable that has different conditional probability mass functions under different stress amplitudes. Then the Lagrange factor polynomial technique is developed to generate the stress-strength interference model with stress-dependent strength. This model assumes that the strength probability mass function is Weibull distributed, while the stress probability mass function is Normal distributed. Accuracy of this method is investigated by an aeroengine bearing cage alloy. Structural reliabilities are computed as 0.796 to 0.986 under several operation modes, which are analyzed by varying the Weibull shape parameter from 1 to 6. Then probability mean function estimated by Lagrange factor polynomial has relatively low errors over most span of the stress dependent strength distribution. With this approach stress-dependent strength reliability of aeroengine structural systems can be established conveniently. Meanwhile the application range of the classical stress-strength interference model can be extended.

The Back Page: My Favorite Lesson: Perfect Sample and Population Mean Formula

2013 ◽  
Vol 107 (4) ◽  
pp. 320
Author(s):  
Clarence W. Lienhard
Keyword(s):  
Random Variable ◽  
Mass Function ◽  
Probability Mass Function ◽  
Discrete Random Variable ◽  
Sample Mean ◽  
Population Mean ◽  
Probability Mass

Typically, students readily understand the concepts of the sample mean and a discrete random variable X with its probability mass function. However, they find the population mean formula unduly difficult to comprehend. Using a perfect sample, the author guides readers to discover the population mean formula from the sample mean.

Maximum Likelihood Set for Estimating a Probability Mass Function

2005 ◽  
Vol 17 (7) ◽  
pp. 1508-1530 ◽  
Author(s):  
Bruno M. Jedynak ◽  
Sanjeev Khudanpur
Keyword(s):  
Maximum Likelihood ◽  
Prior Knowledge ◽  
Empirical Distribution ◽  
Random Variable ◽  
Mass Function ◽  
Small Sample ◽  
Probability Mass Function ◽  
Probability Mass ◽  
Dirichlet Prior ◽  
Probability Simplex

We propose a new method for estimating the probability mass function (pmf) of a discrete and finite random variable from a small sample. We focus on the observed counts—the number of times each value appears in the sample—and define the maximum likelihood set (MLS) as the set of pmfs that put more mass on the observed counts than on any other set of counts possible for the same sample size. We characterize the MLS in detail in this article. We show that the MLS is a diamond-shaped subset of the probability simplex [0, 1]k bounded by at most k × (k − 1) hyper-planes, where k is the number of possible values of the random variable. The MLS always contains the empirical distribution, as well as a family of Bayesian estimators based on a Dirichlet prior, particularly the well-known Laplace estimator. We propose to select from the MLS the pmf that is closest to a fixed pmf that encodes prior knowledge. When using Kullback-Leibler distance for this selection, the optimization problem comprises finding the minimum of a convex function over a domain defined by linear inequalities, for which standard numerical procedures are available. We apply this estimate to language modeling using Zipf's law to encode prior knowledge and show that this method permits obtaining state-of-the-art results while being conceptually simpler than most competing methods.

A study on the injective coloring parameters of certain graphs

2016 ◽  
Vol 08 (03) ◽  
pp. 1650052 ◽  
Author(s):  
N. K. Sudev ◽  
K. P. Chithra ◽  
S. Satheesh ◽  
Johan Kok
Keyword(s):  
Graph Coloring ◽  
Random Variable ◽  
Mass Function ◽  
Probability Mass Function ◽  
Proper Coloring ◽  
Discrete Random Variable ◽  
Injective Coloring ◽  
Random Experiment ◽  
Mean And Variance ◽  
Probability Mass

Coloring the vertices of a graph [Formula: see text] according to certain conditions can be considered as a random experiment and a discrete random variable (r.v.) [Formula: see text] can be defined as the number of vertices having a particular color in the proper coloring of [Formula: see text] and a probability mass function for this random variable can be defined accordingly. In this paper, we extend the concepts of mean and variance to a modified injective graph coloring and determine the values of these parameters for a number of standard graphs.

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