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.

scholarly journals Note on the weighted internal path length of b-ary trees

2007 ◽  
Vol Vol. 9 no. 1 (Analysis of Algorithms) ◽  
Author(s):  
Ludger Rüschendorf ◽  
Eva-Maria Schopp
Keyword(s):  
Limit Theorem ◽  
Continuous Time ◽  
Path Length ◽  
Analysis Of Algorithms ◽  
Random Variables ◽  
Time Parameter ◽  
Recursive Sequences ◽  
International Audience

Analysis of Algorithms International audience In a recent paper Broutin and Devroye (2005) have studied the height of a class of edge-weighted random trees.This is a class of trees growing in continuous time which includes many wellknown trees as examples. In this paper we derive a limit theorem for the internal path length for this class of trees.For the proof we extend a limit theorem in Neininger and Rüschendorf (2004) to recursive sequences of random variables with continuous time parameter.

On characterizations of the geometric distribution by independence of functions of order statistics

1986 ◽  
Vol 23 (1) ◽  
pp. 227-232 ◽  
Author(s):  
R. C. Srivastava
Keyword(s):  
Order Statistics ◽  
Geometric Distribution ◽  
Random Variables ◽  
Weak Form ◽  
Discrete Distributions ◽  
Independence Properties

Let X1, · ··, Xn, n ≧ 2 be i.i.d. random variables having a geometric distribution, and let Y1 ≦ Y2 ≦ · ·· ≦ Yn denote the corresponding order statistics. Define Rn = Yn – Y1 and Zn = Σj=2n (Υj – Y1). Then it is well known that (i) Y, and Rn are independent and (ii) Y1 and Zn are independent. In this paper, we show that a very weak form of each of these independence properties is a characterizing property of the geometric distribution in the class of discrete distributions.

scholarly journals Fitting of a Markov chain model for daily rainfall data at Calcutta

MAUSAM ◽  
2021 ◽  
Vol 22 (1) ◽  
pp. 67-74
Author(s):  
A. N. BASU
Keyword(s):  
Markov Chain ◽  
Geometric Distribution ◽  
Markov Chain Model ◽  
Probability Model ◽  
Daily Rainfall ◽  
Rainfall Data ◽  
Actual Data ◽  
Chain Model ◽  
Daily Rainfall Data ◽  
Rainy Days

A Markov chain probability model has been fitted to the daily rainfall data recorded at Calcutta. The 'wet spell' and 'weather cycles' are found to obey geometric distribution, The distribution of the number of rainy days per week has been calculated and compared with the actual data.

scholarly journals Multiplexing: A

2021 ◽  
pp. 39-58
Author(s):  
Jean Walrand
Keyword(s):  
High Speed ◽  
Probabilistic Models ◽  
Markov Chain Model ◽  
Radio Channel ◽  
Random Variables ◽  
Mean Value ◽  
Chain Model ◽  
Network Provisioning ◽  
Desktop Computers ◽  
Multiple Access Protocol

AbstractThis chapter explores the fluctuations of random variables away from their mean value. You flip a fair coin 100 times. How likely is it that you get 60 heads? Conversely, if you get 60 heads, how likely is it that the coin is fair? Such questions are fundamental in extracting information from data.In Sect. 3.1, we start by exploring the rate available to a user when a random number of them share a link, as illustrated in Sect. 3.1. Such calculations are central to network provisioning. The main analytical tool is the Central Limit Theorem explained in Sect. 3.2 where Gaussian random variables are also introduced and confidence intervals are defined. To share a common link, devices may be attached to a switch. For instance, the desktop computers in a building are typically connected to a switch that then sends the data to a common high-speed link. We explore the delays that packets face through the buffer of a switch in Sect. 3.3. The analysis uses a Markov chain model of the buffer. To share a wireless radio channel, devices use a multiple access protocol that regulates the transmissions. We study such schemes in Sect. 3.4. We use probabilistic models of the protocols.

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 $b \geq 1$.

scholarly journals d-records in geometrically distributed random variables

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Geometric Distribution ◽  
Random Variables ◽  
Random Permutations ◽  
Weak Model ◽  
Asymptotic Formulae ◽  
International Audience

International audience We study d-records in sequences generated by independent geometric random variables and derive explicit and asymptotic formulæ for expectation and variance. Informally speaking, a d-record occurs, when one computes the d-largest values, and the variable maintaining it changes its value while the sequence is scanned from left to right. This is done for the "strict model," but a "weak model" is also briefly investigated. We also discuss the limit q → 1 (q the parameter of the geometric distribution), which leads to the model of random permutations.

On characterizations of the geometric distribution by independence of functions of order statistics

1986 ◽  
Vol 23 (01) ◽  
pp. 227-232
Author(s):  
R. C. Srivastava
Keyword(s):  
Order Statistics ◽  
Geometric Distribution ◽  
Random Variables ◽  
Weak Form ◽  
Discrete Distributions ◽  
Independence Properties

Let X1, · ··, X n , n ≧ 2 be i.i.d. random variables having a geometric distribution, and let Y 1 ≦ Y 2 ≦ · ·· ≦ Y n denote the corresponding order statistics. Define Rn = Yn – Y 1 and Z n = Σ j=2 n (Υ j – Y1). Then it is well known that (i) Y, and Rn are independent and (ii) Y 1 and Zn are independent. In this paper, we show that a very weak form of each of these independence properties is a characterizing property of the geometric distribution in the class of discrete distributions.

An Efficient Markov Chain Model for the Simulation of Heterogeneous Soil Structure

2004 ◽  
Vol 68 (2) ◽  
pp. 346 ◽  
Author(s):  
Keijan Wu ◽  
Naoise Nunan ◽  
John W. Crawford ◽  
Iain M. Young ◽  
Karl Ritz
Keyword(s):  
Markov Chain ◽  
Soil Structure ◽  
Markov Chain Model ◽  
Chain Model ◽  
Heterogeneous Soil
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