A characterization of the Poisson distribution in discrete models with perturbation

1977 ◽  
Vol 5 (10) ◽  
Author(s):  
Lucja Grzegórska
Keyword(s):  
Poisson Distribution ◽  
Discrete Models

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 a characterization of Poisson distributions

1972 ◽  
Vol 9 (4) ◽  
pp. 852-856 ◽  
Author(s):  
J. Aczél
Keyword(s):  
Poisson Distribution ◽  
Random Variables ◽  
Mutual Dependence ◽  
Binomial Distributions ◽  
Destructive Process ◽  
Poisson Distributions ◽  
Bivariate Poisson Distribution

The conjecture pronounced at the end of the paper of Srivastava and Srivastava (1970) is proved in this paper. It gives the following characterization of (bivariate) Poisson distributions. Suppose that items of two types have been observed certain numbers of times, but these original observations have been reduced due to a destructive process which is the product of two binomial distributions and that the probabilities of these reduced numbers are the same whether damaged or undamaged. Then the original random variables had a bivariate Poisson distribution with zero mutual dependence coefficient.

A simultaneous characterization of the Poisson and bernoulli distributions

1981 ◽  
Vol 18 (1) ◽  
pp. 316-320 ◽  
Author(s):  
George Kimeldorf ◽  
Detlef Plachky ◽  
Allan R. Sampson
Keyword(s):  
Poisson Distribution ◽  
Random Variables ◽  
Independent Random Variables ◽  
Constant Independent ◽  
Bernoulli Distribution ◽  
Bernoulli Distributions

Let N, X1, X2, · ·· be non-constant independent random variables with X1, X2, · ·· being identically distributed and N being non-negative and integer-valued. It is shown that the independence of and implies that the Xi's have a Bernoulli distribution and N has a Poisson distribution. Other related characterization results are considered.

An extension of the Rao-Rubin characterization of the Poisson distribution

1977 ◽  
Vol 14 (03) ◽  
pp. 640-646 ◽  
Author(s):  
D. N. Shanbhag
Keyword(s):  
Poisson Distribution ◽  
Present Note ◽  
Related Result ◽  
Renewal Theory ◽  
Extended Version ◽  
Monotonic Functions ◽  
Absolutely Monotonic Functions ◽  
Absolutely Monotonic

Using Bernstein's theorem concerning absolutely monotonic functions, Rao and Rubin (1964) have established a characterization of the Poisson distribution. The present note arrives at an extended version of this result using a technique existing in the renewal theory. A variant of a related result due to Srivastava and Srivastava (1970) and an extension of the result of Talwalker (1970) are also presented.

An elementary proof for the Rao-Rubin characterization of the Poisson distribution

1974 ◽  
Vol 11 (01) ◽  
pp. 211-215 ◽  
Author(s):  
D. N. Shanbhag
Keyword(s):  
Poisson Distribution ◽  
Elementary Proof

This note gives elementary proofs for the characterizations of the Poisson distribution given by Rao and Rubin (1964) and Talwalker (1970) and disproves a conjecture of R. C. and A. B. L. Srivastava (1970).

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