scholarly journals Compositions and samples of geometric random variables with constrained multiplicities

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Margaret Archibald ◽  
Arnold Knopfmacher ◽  
Toufik Mansour
Keyword(s):  
Positive Integer ◽  
Geometric Distribution ◽  
Random Variables ◽  
Related Case ◽  
Forbidden Set ◽  
International Audience ◽  
Nous Examinons ◽  
Independent Identically Distributed

International audience We investigate the probability that a random composition (ordered partition) of the positive integer $n$ has no parts occurring exactly $j$ times, where $j$ belongs to a specified finite $\textit{`forbidden set'}$ $A$ of multiplicities. This probability is also studied in the related case of samples $\Gamma =(\Gamma_1,\Gamma_2,\ldots, \Gamma_n)$ of independent, identically distributed random variables with a geometric distribution. Nous examinons la probabilité qu'une composition faite au hasard (une partition ordonnée) du nombre entier positif $n$ n'a pas de parties qui arrivent exactement $j$ fois, où $j$ appartient à une série interdite, finie et spécifiée $A$ de multiplicités. Cette probabilité est aussi étudiée dans le cas des suites $\Gamma =(\Gamma_1,\Gamma_2,\ldots,\Gamma_n)$ de variables aléatoires identiquement distribuées et indépendantes avec une distribution géométrique.

scholarly journals Samples of geometric random variables with multiplicity constraints

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Margaret Archibald ◽  
Arnold Knopfmacher
Keyword(s):  
Geometric Distribution ◽  
Random Variables ◽  
Fixed Integer ◽  
Forbidden Set ◽  
International Audience ◽  
Independent Identically Distributed

International audience We investigate the probability that a sample $\Gamma=(\Gamma_1,\Gamma_2,\ldots,\Gamma_n)$ of independent, identically distributed random variables with a geometric distribution has no elements occurring exactly $j$ times, where $j$ belongs to a specified finite $\textit{'forbidden set'}$ $A$ of multiplicities. Specific choices of the set $A$ enable one to determine the asymptotic probabilities that such a sample has no variable occuring with multiplicity $b$, or which has all multiplicities greater than $b$, for any fixed integer $b \geq 1$.

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.

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 (2) ◽  
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 X1, X2, …, Xn be independent identically distributed positive integer-valued random variables with order statistics X1:n, X2: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 X1:n–k. Also the random variable X2:n – X1:n is independent of the event [X1:n = 1]. Under mild conditions each of these two properties characterizes the geometric distribution.

A clustering law for some discrete order statistics

2003 ◽  
Vol 40 (01) ◽  
pp. 226-241 ◽  
Author(s):  
Sunder Sethuraman
Keyword(s):  
Distribution Function ◽  
Positive Integer ◽  
Order Statistics ◽  
Law Of Large Numbers ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Large Numbers ◽  
The Law ◽  
Sample Maximum ◽  
Independent Identically Distributed

Let X 1, X 2, …, X n be a sequence of independent, identically distributed positive integer random variables with distribution function F. Anderson (1970) proved a variant of the law of large numbers by showing that the sample maximum moves asymptotically on two values if and only if F satisfies a ‘clustering’ condition, In this article, we generalize Anderson's result and show that it is robust by proving that, for any r ≥ 0, the sample maximum and other extremes asymptotically cluster on r + 2 values if and only if Together with previous work which considered other asymptotic properties of these sample extremes, a more detailed asymptotic clustering structure for discrete order statistics is presented.

scholarly journals An urn model arising from all-optical networks

2006 ◽  
Vol 43 (04) ◽  
pp. 952-966
Author(s):  
John A. Morrison
Keyword(s):  
Positive Integer ◽  
Explicit Expression ◽  
Optical Networks ◽  
Generating Function ◽  
Random Variables ◽  
Urn Model ◽  
Factorial Moments ◽  
Occupancy Model ◽  
All Optical ◽  
Independent Identically Distributed

An occupancy model that has arisen in the investigation of randomized distributed schedules in all-optical networks is considered. The model consists of B initially empty urns, and at stage j of the process d j ≤ B balls are placed in distinct urns with uniform probability. Let M i (j) denote the number of urns containing i balls at the end of stage j. An explicit expression for the joint factorial moments of M 0(j) and M 1(j) is obtained. A multivariate generating function for the joint factorial moments of M i (j), 0 ≤ i ≤ I, is derived (where I is a positive integer). Finally, the case in which the d j , j ≥ 1, are independent, identically distributed random variables is investigated.

A clustering law for some discrete order statistics

2003 ◽  
Vol 40 (1) ◽  
pp. 226-241 ◽  
Author(s):  
Sunder Sethuraman
Keyword(s):  
Distribution Function ◽  
Positive Integer ◽  
Order Statistics ◽  
Law Of Large Numbers ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Large Numbers ◽  
The Law ◽  
Sample Maximum ◽  
Independent Identically Distributed

Let X1, X2, …, Xn be a sequence of independent, identically distributed positive integer random variables with distribution function F. Anderson (1970) proved a variant of the law of large numbers by showing that the sample maximum moves asymptotically on two values if and only if F satisfies a ‘clustering’ condition, In this article, we generalize Anderson's result and show that it is robust by proving that, for any r ≥ 0, the sample maximum and other extremes asymptotically cluster on r + 2 values if and only if Together with previous work which considered other asymptotic properties of these sample extremes, a more detailed asymptotic clustering structure for discrete order statistics is presented.

scholarly journals A combinatorial and probabilistic study of initial and end heights of descents in samples of geometrically distributed random variables and in permutations

2007 ◽  
Vol Vol. 9 no. 1 (Analysis of Algorithms) ◽  
Author(s):  
Guy Louchard ◽  
Helmut Prodinger
Keyword(s):  
Geometric Distribution ◽  
Analysis Of Algorithms ◽  
Markov Chain Model ◽  
Random Variables ◽  
Discrete Distributions ◽  
Chain Model ◽  
Initial Height ◽  
Probabilistic Study ◽  
International Audience ◽  
Expected Values

Analysis of Algorithms International audience In words, generated by independent geometrically distributed random variables, we study the lth descent, which is, roughly speaking, the lth occurrence of a neighbouring pair ab with a>b. The value a is called the initial height, and b the end height. We study these two random variables (and some similar ones) by combinatorial and probabilistic tools. We find in all instances a generating function Ψ(v,u), where the coefficient of vjui refers to the jth descent (ascent), and i to the initial (end) height. From this, various conclusions can be drawn, in particular expected values. In the probabilistic part, a Markov chain model is used, which allows to get explicit expressions for the heights of the second descent. In principle, one could go further, but the complexity of the results forbids it. This is extended to permutations of a large number of elements. Methods from q-analysis are used to simplify the expressions. This is the reason that we confine ourselves to the geometric distribution only. For general discrete distributions, no such tools are available.

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.

Sign in / Sign up

Export Citation Format

Share Document