A note on the generalized Rao–Rubin condition and characterization of certain discrete distributions

1980 ◽  
Vol 17 (02) ◽  
pp. 563-569
Author(s):  
Sheela Talwalker
Keyword(s):  
Discrete Distribution ◽  
Random Variable ◽  
Discrete Distributions ◽  
Binomial Probability ◽  
Poisson Distributions ◽  
Image Position ◽  
Original Observation

Assuming that the original observation from a discrete distribution is subject to damage according to a binomial probability law, it is shown here that a class of discrete distributions consisting of binomial, Poisson and mixed Poisson distributions, is characterized by the generalized Rao–Rubin condition 0 <π, π′ < 1 are constants. Y denotes the resulting or observed random variable, taking the values r = 0, 1, 2, ….

A note on the generalized Rao–Rubin condition and characterization of certain discrete distributions

1980 ◽  
Vol 17 (2) ◽  
pp. 563-569 ◽  
Author(s):  
Sheela Talwalker
Keyword(s):  
Discrete Distribution ◽  
Random Variable ◽  
Discrete Distributions ◽  
Binomial Probability ◽  
Poisson Distributions ◽  
Image Position ◽  
Original Observation

Assuming that the original observation from a discrete distribution is subject to damage according to a binomial probability law, it is shown here that a class of discrete distributions consisting of binomial, Poisson and mixed Poisson distributions, is characterized by the generalized Rao–Rubin condition 0 <π, π′ < 1 are constants. Y denotes the resulting or observed random variable, taking the values r = 0, 1, 2, ….

On a characterization of Poisson distribution

1970 ◽  
Vol 7 (2) ◽  
pp. 497-501 ◽  
Author(s):  
R. C. Srivastava ◽  
A. B. L. Srivastava
Keyword(s):  
Poisson Distribution ◽  
Random Variable ◽  
Discrete Models ◽  
Destructive Process ◽  
Image Position ◽  
Original Observation

Recently Rao (1963) has considered discrete models where an original observation produced by nature is subjected to a destructive process and we observe the undestroyed part of the original observation. Suppose the original observation produced by nature is distributed according to a Poisson distribution with parameter λ and the probability that the original observation n is reduced to r due to the destructive process is Now if Y denotes the resulting random variable (r.v.), then it is easily seen that Let us call this condition the *-condition. Later Rao and Rubin ({1964),Theorem 1) proved that the *-condition is a characterizing property of the Poisson distribution.

On a characterization of Poisson distribution

1970 ◽  
Vol 7 (02) ◽  
pp. 497-501 ◽  
Author(s):  
R. C. Srivastava ◽  
A. B. L. Srivastava
Keyword(s):  
Poisson Distribution ◽  
Random Variable ◽  
Discrete Models ◽  
Destructive Process ◽  
Image Position ◽  
Original Observation

Recently Rao (1963) has considered discrete models where an original observation produced by nature is subjected to a destructive process and we observe the undestroyed part of the original observation. Suppose the original observation produced by nature is distributed according to a Poisson distribution with parameter λ and the probability that the original observation n is reduced to r due to the destructive process is Now if Y denotes the resulting random variable (r.v.), then it is easily seen that Let us call this condition the *-condition. Later Rao and Rubin ({1964),Theorem 1) proved that the *-condition is a characterizing property of the Poisson distribution.

On the Distribution of Sum of Independent Positive Binomial Variables

1970 ◽  
Vol 13 (1) ◽  
pp. 151-152 ◽  
Author(s):  
J. C. Ahuja
Keyword(s):  
Power Series ◽  
Inversion Formula ◽  
Probability Function ◽  
Random Variables ◽  
Random Variable ◽  
Characteristic Functions ◽  
Binomial Probability ◽  
Generalized Power Series ◽  
Alternative Derivation ◽  
Image Position

Let X1, X2, …, Xn be n independent and identically distributed random variables having the positive binomial probability function1where 0 < p < 1, and T = {1, 2, …, N}. Define their sum as Y=X1 + X2 + … +Xn. The distribution of the random variable Y has been obtained by Malik [2] using the inversion formula for characteristic functions. It appears that his result needs some correction. The purpose of this note is to give an alternative derivation of the distribution of Y by applying one of the results, established by Patil [3], for the generalized power series distribution.

Characterization of matrix probability distributions by mean residual lifetime

1986 ◽  
Vol 100 (3) ◽  
pp. 583-589
Author(s):  
P. E. Jupp
Keyword(s):  
Symmetric Matrix ◽  
Probability Distributions ◽  
Random Variable ◽  
Residual Lifetime ◽  
Regularity Conditions ◽  
Mean Residual Lifetime ◽  
The Mean ◽  
Image Position ◽  
Vector Valued

The mean residual lifetime of a real-valued random variable X is the function e defined byOne of the more important properties of the mean residual lifetime function is that it determines the distribution of X. See, for example, Swartz [10]. References to related characterizations are given by Galambos and Kotz [3], pages 30–35. It was established by Jupp and Mardia[6] that this property holds also for vector-valued X. As (1·1) makes sense if X is a random symmetric matrix, it is natural to ask whether the property holds in this case also. The purpose of this note is to show that, under certain regularity conditions, the distributions of such matrices are indeed determined by their mean residual lifetimes.

On an Integral Equation for Discounted Compound – Annuity Distributions

1989 ◽  
Vol 19 (2) ◽  
pp. 191-198 ◽  
Author(s):  
Colin M. Ramsay
Keyword(s):  
Integral Equation ◽  
Random Variables ◽  
Random Variable ◽  
Discrete Distributions ◽  
Discount Factor ◽  
Present Value ◽  
New Class ◽  
Image Position ◽  
Independent And Identically Distributed

AbstractWe consider a risk generating claims for a period of N consecutive years (after which it expires), N being an integer valued random variable. Let Xk denote the total claims generated in the kth year, k ≥ 1. The Xk's are assumed to be independent and identically distributed random variables, and are paid at the end of the year. The aggregate discounted claims generated by the risk until it expires is defined as where υ is the discount factor. An integral equation similar to that given by Panjer (1981) is developed for the pdf of SN(υ). This is accomplished by assuming that N belongs to a new class of discrete distributions called annuity distributions. The probabilities in annuity distributions satisfy the following recursion:where an is the present value of an n-year immediate annuity.

scholarly journals Asymptotic results for multiplexing subexponential on-off processes

1999 ◽  
Vol 31 (02) ◽  
pp. 394-421 ◽  
Author(s):  
Predrag R. Jelenković ◽  
Aurel A. Lazar
Keyword(s):  
Queueing System ◽  
Random Variable ◽  
Mean Value ◽  
Activity Period ◽  
Arrival Process ◽  
Asymptotic Results ◽  
Fluid Queue ◽  
Arrival Processes ◽  
Image Position

Consider an aggregate arrival process A N obtained by multiplexing N on-off processes with exponential off periods of rate λ and subexponential on periods τon. As N goes to infinity, with λN → Λ, A N approaches an M/G/∞ type process. Both for finite and infinite N, we obtain the asymptotic characterization of the arrival process activity period. Using these results we investigate a fluid queue with the limiting M/G/∞ arrival process A t ∞ and capacity c. When on periods are regularly varying (with non-integer exponent), we derive a precise asymptotic behavior of the queue length random variable Q t P observed at the beginning of the arrival process activity periods where ρ = 𝔼A t ∞ &lt; c; r (c ≤ r) is the rate at which the fluid is arriving during an on period. The asymptotic (time average) queue distribution lower bound is obtained under more general assumptions regarding on periods than regular variation. In addition, we analyse a queueing system in which one on-off process, whose on period belongs to a subclass of subexponential distributions, is multiplexed with independent exponential processes with aggregate expected rate 𝔼e t . This system is shown to be asymptotically equivalent to the same queueing system with the exponential arrival processes being replaced by their total mean value 𝔼e t .

scholarly journals Asymptotic results for multiplexing subexponential on-off processes

1999 ◽  
Vol 31 (2) ◽  
pp. 394-421 ◽  
Author(s):  
Predrag R. Jelenković ◽  
Aurel A. Lazar
Keyword(s):  
Queueing System ◽  
Random Variable ◽  
Mean Value ◽  
Activity Period ◽  
Arrival Process ◽  
Asymptotic Results ◽  
Fluid Queue ◽  
Arrival Processes ◽  
Image Position

Consider an aggregate arrival process AN obtained by multiplexing N on-off processes with exponential off periods of rate λ and subexponential on periods τon. As N goes to infinity, with λN → Λ, AN approaches an M/G/∞ type process. Both for finite and infinite N, we obtain the asymptotic characterization of the arrival process activity period.Using these results we investigate a fluid queue with the limiting M/G/∞ arrival process At∞ and capacity c. When on periods are regularly varying (with non-integer exponent), we derive a precise asymptotic behavior of the queue length random variable QtP observed at the beginning of the arrival process activity periods where ρ = 𝔼At∞ < c; r (c ≤ r) is the rate at which the fluid is arriving during an on period. The asymptotic (time average) queue distribution lower bound is obtained under more general assumptions regarding on periods than regular variation.In addition, we analyse a queueing system in which one on-off process, whose on period belongs to a subclass of subexponential distributions, is multiplexed with independent exponential processes with aggregate expected rate 𝔼et. This system is shown to be asymptotically equivalent to the same queueing system with the exponential arrival processes being replaced by their total mean value 𝔼et.

scholarly journals Fitting Discrete Distributions on the First Two Moments

1995 ◽  
Vol 9 (4) ◽  
pp. 623-632 ◽  
Author(s):  
Ivo Adan ◽  
Michel van Eenige ◽  
Jacques Resing
Keyword(s):  
Service Time ◽  
Iteration Method ◽  
Time Distribution ◽  
Discrete Distribution ◽  
Random Variable ◽  
Numerical Results ◽  
Discrete Distributions ◽  
Service Time Distribution ◽  
Excellent Performance ◽  
Simple Method

In this paper we present a simple method to fit a discrete distribution on the first two moments of a given random variable. With the Fitted distribution we solve approximately Lindley's equation for the D/G/1 queue with discrete service-time distribution using a moment-iteration method. Numerical results show excellent performance of the method.

Characterization of the logistic and loglogistic distributions by extreme value related stability with random sample size

1987 ◽  
Vol 24 (04) ◽  
pp. 838-851 ◽  
Author(s):  
W. J. Voorn
Keyword(s):  
Positive Integer ◽  
Sample Size ◽  
Random Variable ◽  
Logistic Distribution ◽  
Size Distributions ◽  
Random Sample Size ◽  
Maximum Value ◽  
Maximum Stability ◽  
Independent Observations

Maximum stability of a distribution with respect to a positive integer random variable N is defined by the property that the type of distribution is not changed when considering the maximum value of N independent observations. The logistic distribution is proved to be the only symmetric distribution which is maximum stable with respect to each member of a sequence of positive integer random variables assuming value 1 with probability tending to 1. If a distribution is maximum stable with respect to such a sequence and minimum stable with respect to another, then it must be logistic, loglogistic or ‘backward' loglogistic. The only possible sample size distributions in these cases are geometric.

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