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< ∞.

A Note on the Distribution of the Time of the First k-Record Index

1998 ◽  
Vol 12 (3) ◽  
pp. 321-323
Author(s):  
Mitsushi Tamaki
Keyword(s):  
Generating Function ◽  
Probability Generating Function ◽  
Random Variables ◽  
Mass Function ◽  
Probability Mass Function ◽  
Factorial Moments ◽  
Finite Set ◽  
Probability Mass ◽  
Independent And Identically Distributed

We explicitly give the probability mass function and the probability generating function of the first k-record index for a sequence of independent and identically distributed random variables that take on a finite set of possible values. We also compute its factorial moments.

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.

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 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 Asymptotic expansions for distribution of sums quasi-lattice random variables

2011 ◽  
Vol 52 ◽  
pp. 359-364
Author(s):  
Algimantas Bikelis ◽  
Kazimieras Padvelskis ◽  
Pranas Vaitkus
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Random Variables ◽  
Independent Random Variables

Althoug Chebyshev [3] and Edeworth [5] had conceived of the formal expansions for distribution of sums of independent random variables, but only in Cramer’s work [4] was laid a proper foundation of this problem. In the case when random variables are lattice Esseen get the asymptotic expansion in a new different form. Here we extend this problem for quasi-lattice random variables.  

scholarly journals Uniform renewal theory with applications to expansions of random geometric sums

2007 ◽  
Vol 39 (4) ◽  
pp. 1070-1097 ◽  
Author(s):  
J. Blanchet ◽  
P. Glynn
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Random Variables ◽  
Renewal Theory ◽  
Random Variable ◽  
Higher Order ◽  
Order Correction ◽  
Diffusion Approximations ◽  
Geometric Sums ◽  
Correction Terms

Consider a sequence X = (Xn: n ≥ 1) of independent and identically distributed random variables, and an independent geometrically distributed random variable M with parameter p. The random variable SM = X1 + ∙ ∙ ∙ + XM is called a geometric sum. In this paper we obtain asymptotic expansions for the distribution of SM as p ↘ 0. If EX1 > 0, the asymptotic expansion is developed in powers of p and it provides higher-order correction terms to Renyi's theorem, which states that P(pSM > x) ≈ exp(-x/EX1). Conversely, if EX1 = 0 then the expansion is given in powers of √p. We apply the results to obtain corrected diffusion approximations for the M/G/1 queue. These expansions follow in a unified way as a consequence of new uniform renewal theory results that are also developed in this paper.

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