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 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 A finite form for the wrapped Poisson distribution

1992 ◽  
Vol 24 (01) ◽  
pp. 221-222 ◽  
Author(s):  
Frank Ball ◽  
Paul Blackwell
Keyword(s):  
Poisson Distribution ◽  
Mass Function ◽  
Probability Mass Function ◽  
Finite Form ◽  
Probabilistic Proof ◽  
Probability Mass

We give a finite form for the probability mass function of the wrapped Poisson distribution, together with a probabilistic proof. We also describe briefly its connection with existing results.

Generalized Hypergeometric Factorial Moment Distribution of Order k

2000 ◽  
Vol 50 (1-2) ◽  
pp. 71-78 ◽  
Author(s):  
C. Satheesh Kumar ◽  
T. S. K. Moothathu
Keyword(s):  
Negative Binomial ◽  
Factorial Moment ◽  
Mass Function ◽  
Discrete Distributions ◽  
Probability Mass Function ◽  
Factorial Moments ◽  
Point Distribution ◽  
Moment Distribution ◽  
Special Cases ◽  
Probability Mass

Here we introduce the generalized hypergeometric functional moment distribution of order k (GHFMD (k)) in the distribution of the random sum [Formula: see text] having Hira no's k-point distribution, where N, independent of X j's, has the generalized hypergeomet ric factorial moment distribution. Well-known discrete distributions of order k such as cluster binomial, cluster negative binomial and extended Poisson are shown to be special cases of GHFMD(k). The probability mass function, recurrence relations for probabilities and factorial moments of GHFMD (k) are found out. The beta or the gamma mixture of GHFMD (k) is shown to be a GHFMD (k). Finally GHFMD (k) is obtained as a limit of another GHFMD (k). AMS (2000) Subject Classification: Primary 60E05, 60E10; Secondary 33C20.

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 A finite form for the wrapped Poisson distribution

1992 ◽  
Vol 24 (1) ◽  
pp. 221-222 ◽  
Author(s):  
Frank Ball ◽  
Paul Blackwell
Keyword(s):  
Poisson Distribution ◽  
Mass Function ◽  
Probability Mass Function ◽  
Finite Form ◽  
Probabilistic Proof ◽  
Probability Mass

We give a finite form for the probability mass function of the wrapped Poisson distribution, together with a probabilistic proof. We also describe briefly its connection with existing results.

scholarly journals The Novel Bivariate Distribution: Statistical Properties and Real Data Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
B. I. Mohammed ◽  
Abdulaziz S. Alghamdi ◽  
Hassan M. Aljohani ◽  
Md. Moyazzem Hossain
Keyword(s):  
Negative Binomial ◽  
Real Data ◽  
Information Criterion ◽  
Mass Function ◽  
Bivariate Distribution ◽  
Statistical Properties ◽  
Model Parameters ◽  
Probability Mass Function ◽  
New Class ◽  
Probability Mass

This article proposes a novel class of bivariate distributions that are completely defined by stating their conditionals as Poisson exponential distributions. Numerous statistical properties of this distribution are also examined here, including the conditional probability mass function (PMF) and moments of the new class. The techniques of maximum likelihood and pseudolikelihood are used to estimate the model parameters. Additionally, the effectiveness of the bivariate Poisson exponential conditional (BPEC) distribution is compared to that of the bivariate Poisson conditional (BPC), the bivariate Poisson (BP), the bivariate Poisson–Lindley (BPL), and the bivariate negative binomial (BNB) distributions using a real-world dataset. The findings of Akaike information criterion (AIC) and Bayesian information criterion (BIC) reveal that the BPEC distribution performs better than the other distributions considered in this study. As a result, the authors claim that this distribution may be used to fit dependent and overspread count data.

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.

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