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.

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.

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 characterization of the negative exponential distribution with application to reliability theory

1981 ◽  
Vol 18 (3) ◽  
pp. 652-659 ◽  
Author(s):  
M. J. Phillips
Keyword(s):  
Exponential Distribution ◽  
Random Variables ◽  
Reliability Theory ◽  
The Other ◽  
Independent Random Variables ◽  
System Availability ◽  
Failure Times ◽  
Negative Exponential Distribution ◽  
Negative Exponential

The negative exponential distribution is characterized in terms of two independent random variables. Only one of the random variables has a negative exponential distribution whilst the other can belong to a wide class of distributions. This result is then applied to two models for the reliability of a system of two modules subject to revealed and unrevealed faults to show when the models are equivalent. It is also shown, under certain conditions, that the system availability is only independent of the distribution of revealed failure times in one module when unrevealed failure times in the other module have a negative exponential distribution.

A joint characterization of the multinomial distribution and the Poisson process

1983 ◽  
Vol 20 (01) ◽  
pp. 202-208 ◽  
Author(s):  
George Kimeldorf ◽  
Peter F. Thall
Keyword(s):  
Poisson Process ◽  
Present Article ◽  
Point Process ◽  
Random Variables ◽  
Multinomial Distribution ◽  
Process Theory ◽  
Independent Random Variables ◽  
Mutually Independent

It has been recently proved that if N, X 1, X 2, … are non-constant mutually independent random variables with X 1,X 2, … identically distributed and N non-negative and integer-valued, then the independence of and implies that X 1 is Bernoulli and N is Poisson. A well-known theorem in point process theory due to Fichtner characterizes a Poisson process in terms of a sum of independent thinnings. In the present article, simultaneous generalizations of both of these results are provided, including a joint characterization of the multinomial distribution and the Poisson process.

A characterization of the negative exponential distribution with application to reliability theory

1981 ◽  
Vol 18 (03) ◽  
pp. 652-659 ◽  
Author(s):  
M. J. Phillips
Keyword(s):  
Exponential Distribution ◽  
Random Variables ◽  
Reliability Theory ◽  
The Other ◽  
Independent Random Variables ◽  
System Availability ◽  
Failure Times ◽  
Negative Exponential Distribution ◽  
Negative Exponential

The negative exponential distribution is characterized in terms of two independent random variables. Only one of the random variables has a negative exponential distribution whilst the other can belong to a wide class of distributions. This result is then applied to two models for the reliability of a system of two modules subject to revealed and unrevealed faults to show when the models are equivalent. It is also shown, under certain conditions, that the system availability is only independent of the distribution of revealed failure times in one module when unrevealed failure times in the other module have a negative exponential distribution.

scholarly journals On Error Bounds for Approximations to Multivariate Distributions II

2002 ◽  
Vol 32 (1) ◽  
pp. 57-69
Author(s):  
Bjørn Sundt ◽  
Raluca Vernic
Keyword(s):  
Error Bounds ◽  
Random Variables ◽  
Independent Random Variables ◽  
Multivariate Distributions ◽  
Dependent Random Variables ◽  
Linear Transforms ◽  
Original Distribution ◽  
Multivariate Bernoulli ◽  
Bernoulli Distributions

AbstractIn the present paper, we study error bounds for approximations to multivariate distributions. In particular, we discuss some general versions of compound multivariate distributions and look at distributions of dependent random variables constructed by linear transforms of independent random variables or vectors. Special attention is paid to the case when the support of the original distribution is restricted. We also look at some applications with multivariate Bernoulli distributions.

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 joint characterization of the multinomial distribution and the Poisson process

1983 ◽  
Vol 20 (1) ◽  
pp. 202-208 ◽  
Author(s):  
George Kimeldorf ◽  
Peter F. Thall
Keyword(s):  
Poisson Process ◽  
Present Article ◽  
Point Process ◽  
Random Variables ◽  
Multinomial Distribution ◽  
Process Theory ◽  
Independent Random Variables ◽  
Mutually Independent

It has been recently proved that if N, X1, X2, … are non-constant mutually independent random variables with X1,X2, … identically distributed and N non-negative and integer-valued, then the independence of and implies that X1 is Bernoulli and N is Poisson. A well-known theorem in point process theory due to Fichtner characterizes a Poisson process in terms of a sum of independent thinnings. In the present article, simultaneous generalizations of both of these results are provided, including a joint characterization of the multinomial distribution and the Poisson process.

Characterization of a class of second-order density functions

1967 ◽  
Vol 4 (1) ◽  
pp. 123-129 ◽  
Author(s):  
C. B. Mehr
Keyword(s):  
Exponential Distribution ◽  
Gamma Distribution ◽  
Gaussian Distribution ◽  
Random Variables ◽  
Second Order ◽  
Independent Random Variables ◽  
Density Functions ◽  
Linear Filtering ◽  
Non Linear

Distributions of some random variables have been characterized by independence of certain functions of these random variables. For example, let X and Y be two independent and identically distributed random variables having the gamma distribution. Laha showed that U = X + Y and V = X | Y are also independent random variables. Lukacs showed that U and V are independently distributed if, and only if, X and Y have the gamma distribution. Ferguson characterized the exponential distribution in terms of the independence of X – Y and min (X, Y). The best-known of these characterizations is that first proved by Kac which states that if random variables X and Y are independent, then X + Y and X – Y are independent if, and only if, X and Y are jointly Gaussian with the same variance. In this paper, Kac's hypotheses have been somewhat modified. In so doing, we obtain a larger class of distributions which we shall call class λ1. A subclass λ0 of λ1 enjoys many nice properties of the Gaussian distribution, in particular, in non-linear filtering.

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