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.

On a characterization of Poisson distributions

1972 ◽  
Vol 9 (04) ◽  
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.

Another characteristic property of the Poisson distribution

1970 ◽  
Vol 68 (1) ◽  
pp. 167-169 ◽  
Author(s):  
D. N. Shanbhag
Keyword(s):  
Poisson Distribution ◽  
Conditional Distribution ◽  
Characteristic Property ◽  
Random Variables ◽  
Independent Random Variables ◽  
Poisson Distributions

1. Introduction: In (4) Moran considers two independent random variables X and Y taking non-negative integral values to give a characterization of the Poisson distribution. He establishes that the conditional distribution of X, given the total X + Y, is binomial for all given values of X + Y and there exists at least one i so that P(x = i) > 0, P( Y = i) > 0 if and only if X and Y have Poisson distributions. A slightly improved version of this result is given by Chatterji (1). For a comprehensive bibliography on the Poisson distribution the reader is referred to (3).

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.

scholarly journals Compound bi-free Poisson distributions

2019 ◽  
Vol 22 (02) ◽  
pp. 1950014
Author(s):  
Mingchu Gao
Keyword(s):  
Poisson Distribution ◽  
Matrix Model ◽  
Random Variables ◽  
Divisible Distribution ◽  
Infinitely Divisible Distribution ◽  
Infinitely Divisible ◽  
Random Matrix Model ◽  
Poisson Distributions ◽  
Poisson Limit ◽  
The Right

In this paper, we study compound bi-free Poisson distributions for two-faced families of random variables. We prove a Poisson limit theorem for compound bi-free Poisson distributions. Furthermore, a bi-free infinitely divisible distribution for a two-faced family of self-adjoint random variables can be realized as the limit of a sequence of compound bi-free Poisson distributions of two-faced families of self-adjoint random variables. If a compound bi-free Poisson distribution is determined by a positive number and the distribution of a two-faced family of finitely many random variables, which has an almost sure random matrix model, and the left random variables commute with the right random variables in the two-faced family, then we can construct a random bi-matrix model for the compound bi-free Poisson distribution. If a compound bi-free Poisson distribution is determined by a positive number and the distribution of a commutative pair of random variables, we can construct an asymptotic bi-matrix model with entries of creation and annihilation operators for the compound bi-free Poisson distribution.

A simultaneous characterization of the Poisson and bernoulli distributions

1981 ◽  
Vol 18 (01) ◽  
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, X 1, X 2, · ·· be non-constant independent random variables with X 1, X 2, · ·· 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.

scholarly journals A Note on the (Weighted) Bivariate Poisson Distribution

2021 ◽  
Vol 14 (1) ◽  
pp. 192-203
Author(s):  
R. Bidounga ◽  
P. C. Batsindila Nganga ◽  
L. Niéré ◽  
D. Mizère
Keyword(s):  
Poisson Distribution ◽  
Poisson Distributions ◽  
Standard Distribution ◽  
Bivariate Poisson Distribution

In the recent statistical literature, the univariate Poisson distribution has been generalized by many authors, among them: the univariate weighted Poisson distribution [13], the generalized univariate Poisson distribution [7], the bivariate Poisson distribution according to Holgate [11], the bivariate Poisson distribution according to Lakshminarayana, Pandit and Srinivasa Rao [15], the bivariate Poisson distribution according to Berkhout and Plug [4], the bivariate weighted Poisson distribution according to Elion et al. [8] and the generalized bivariate Poisson distribution according to Famoye [9]. In this paper, We highlight the weighted bivariate Poisson distribution and show that it is the synthesis of all the bivariate Poisson distributions which, under certain conditions, converge in distribution towards the bivariate Poisson distribution according to Berkhout and Plug [4] which can be considered like the standard distribution in N2 as is the univariate Poisson distribution in N.

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.

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