scholarly journals Characterization of the Geometric Distribution Via Linear Combinations of Observations and of Records

Sankhya A ◽  
2021 ◽  
Author(s):  
Barry C. Arnold ◽  
Jose A. Villasenor
Keyword(s):  
Linear Combination ◽  
Geometric Distribution ◽  
Random Variables ◽  
Record Values ◽  
Linear Combinations ◽  
Distributional Property ◽  
Weak Records ◽  
Geometric Variables ◽  
Independent Identically Distributed

AbstractIn a sequence of independent identically distributed geometric random variables, the sum of the first two record values is distributed as a simple linear combination of geometric variables. It is verified that this distributional property characterizes the geometric distribution. A related characterization conjecture is also discussed. Related discussion in the context of weak records is also provided.

Weak Records of Geometric Distribution

2005 ◽  
Vol 56 (1-4) ◽  
pp. 295-304
Author(s):  
M. Ahsanullah
Keyword(s):  
Geometric Distribution ◽  
Random Variables ◽  
Record Values ◽  
Distributional Properties ◽  
Weak Records ◽  
Independent Identically Distributed

Summary Upper weak record values from a sequence of independent identically distributed random variables are considered. Several distributional properties of the upper weak records from geometric distribution are presented. Based on these distributional properties some characterizations of the geometric distribution are given.

A characterization of the geometric distribution

1983 ◽  
Vol 20 (01) ◽  
pp. 209-212 ◽  
Author(s):  
M. Sreehari
Keyword(s):  
Positive Integer ◽  
Order Statistics ◽  
Geometric Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Characterization Problem ◽  
Independent Identically Distributed

Let X 1, X 2, …, Xn be independent identically distributed positive integer-valued random variables with order statistics X 1:n , X 2:n , …, X n:n . We prove that if the random variable X2:n – X 1:n is independent of the events [X1:n = m] and [X1:n = k], for fixed k > m > 1, then the Xi 's are geometric. This is related to a characterization problem raised by Arnold (1980).

A characterization of the geometric distribution

1983 ◽  
Vol 20 (1) ◽  
pp. 209-212 ◽  
Author(s):  
M. Sreehari
Keyword(s):  
Positive Integer ◽  
Order Statistics ◽  
Geometric Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Characterization Problem ◽  
Independent Identically Distributed

Let X1, X2, …, Xn be independent identically distributed positive integer-valued random variables with order statistics X1:n, X2:n, …, Xn:n. We prove that if the random variable X2:n – X1:n is independent of the events [X1:n = m] and [X1:n = k], for fixed k > m > 1, then the Xi's are geometric. This is related to a characterization problem raised by Arnold (1980).

Two characterizations of the geometric distribution

1980 ◽  
Vol 17 (02) ◽  
pp. 570-573 ◽  
Author(s):  
Barry C. Arnold
Keyword(s):  
Positive Integer ◽  
Order Statistics ◽  
Conditional Distribution ◽  
Geometric Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Mild Conditions ◽  
Independent Identically Distributed

Let X 1, X 2, …, Xn be independent identically distributed positive integer-valued random variables with order statistics X 1:n , X 2:n , …, Xn :n. If the Xi 's have a geometric distribution then the conditional distribution of Xk +1:n – Xk :n given Xk+ 1:n – Xk :n > 0 is the same as the distribution of X 1:n–k . Also the random variable X 2:n – X 1:n is independent of the event [X 1:n = 1]. Under mild conditions each of these two properties characterizes the geometric distribution.

EQUIVALENT CHARACTERIZATIONS ON ORDERINGS OF ORDER STATISTICS AND SAMPLE RANGES

2010 ◽  
Vol 24 (2) ◽  
pp. 245-262 ◽  
Author(s):  
Tiantian Mao ◽  
Taizhong Hu
Keyword(s):  
Order Statistics ◽  
Exponential Distribution ◽  
Geometric Distribution ◽  
Random Variables ◽  
The Other ◽  
Stochastic Orders ◽  
Exponential Random Variables ◽  
Discrete Counterpart ◽  
Similarities And Differences ◽  
Geometric Variables

The purpose of this article is to present several equivalent characterizations of comparing the largest-order statistics and sample ranges of two sets of n independent exponential random variables with respect to different stochastic orders, where the random variables in one set are heterogeneous and the random variables in the other set are identically distributed. The main results complement and extend several known results in the literature. The geometric distribution can be regarded as the discrete counterpart of the exponential distribution. We also study the orderings of the largest-order statistics from geometric random variables and point out similarities and differences between orderings of the largest-order statistics from geometric variables and from exponential variables.

A Characterization of Geometric Distribution Throughkth Weak Records

2005 ◽  
Vol 34 (12) ◽  
pp. 2345-2351 ◽  
Author(s):  
Anna Dembińska ◽  
Fernando López-blázquez
Keyword(s):  
Geometric Distribution ◽  
Weak Records

scholarly journals Some Characterizations of Exponential Distribution

2017 ◽  
Vol 6 (5) ◽  
pp. 132
Author(s):  
M. Ahsanullah ◽  
M. Z. Anis
Keyword(s):  
Distribution Function ◽  
Linear Combination ◽  
Lebesgue Measure ◽  
Exponential Distribution ◽  
Random Variables ◽  
Absolutely Continuous ◽  
Measure Distribution ◽  
Independent And Identically Distributed

There are some characterizations of the exponential distribution based on the relation of the maximum of two observations expressed as linear combination of the two observations. In this paper some generalizations of this known characterization of the exponential distribution using the relations between the maximum and minimum of  independent and identically distributed random variables having absolutely continuous (with respect to Lebesgue measure) distribution function will be presented.

scholarly journals A Geometric Derivation of the Irwin-Hall Distribution

2017 ◽  
Vol 2017 ◽  
pp. 1-6 ◽  
Author(s):  
James E. Marengo ◽  
David L. Farnsworth ◽  
Lucas Stefanic
Keyword(s):  
Finite Number ◽  
Finite Length ◽  
Random Variables ◽  
Unit Interval ◽  
Exclusion Principle ◽  
Linear Combinations ◽  
Normal Distributions ◽  
Sums Of Random Variables ◽  
Special Cases ◽  
Independent Identically Distributed

The Irwin-Hall distribution is the distribution of the sum of a finite number of independent identically distributed uniform random variables on the unit interval. Many applications arise since round-off errors have a transformed Irwin-Hall distribution and the distribution supplies spline approximations to normal distributions. We review some of the distribution’s history. The present derivation is very transparent, since it is geometric and explicitly uses the inclusion-exclusion principle. In certain special cases, the derivation can be extended to linear combinations of independent uniform random variables on other intervals of finite length. The derivation adds to the literature about methodologies for finding distributions of sums of random variables, especially distributions that have domains with boundaries so that the inclusion-exclusion principle might be employed.

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