Distributions of Patterns of Pair of Successes Separated by Failure Runs of Length at Least and at Most Involving Markov Dependent Trials: GERT Approach

doi:10.1155/2013/494976

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Distributions of Patterns of Pair of Successes Separated by Failure Runs of Length at Least and at Most Involving Markov Dependent Trials: GERT Approach

Journal Article•DOI•

Distributions of Patterns of Pair of Successes Separated by Failure Runs of Length at Least and at Most Involving Markov Dependent Trials: GERT Approach

Kanwar Sen¹, Pooja Mohan, Manju Agarwal²•Institutions (2)

University of Delhi¹, Shiv Nadar University²

05 Jun 2013-Vol. 2013, pp 1-9

TL;DR: In this article, the authors used the Graphical Evaluation and Review Technique (GERT) to obtain probability generating functions of the waiting time distributions of 1st, and th nonoverlapping and overlapping occurrences of the pattern, involving homogenous Markov dependent trials.

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Abstract: We use the Graphical Evaluation and Review Technique (GERT) to obtain probability generating functions of the waiting time distributions of 1st, and th nonoverlapping and overlapping occurrences of the pattern , involving homogenous Markov dependent trials. GERT besides providing visual picture of the system helps to analyze the system in a less inductive manner. Mean and variance of the waiting times of the occurrence of the patterns have also been obtained. Some earlier results existing in literature have been shown to be particular cases of these results.

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31,532 citations

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An Introduction to Probability

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Lawrence N. Dworsky

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Journal Article•DOI•

Distribution Theory of Runs: A Markov Chain Approach

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James C. Fu¹, Markos V. Koutras¹•Institutions (1)

University of Manitoba¹

01 Sep 1994-Journal of the American Statistical Association

TL;DR: In this article, a unified approach for the distribution theory of runs based on a finite Markov chain imbedding technique is presented, which covers both identical and non-identical Bernoulli trials.

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Abstract: The statistics of the number of success runs in a sequence of Bernoulli trials have been used in many statistical areas. For almost a century, even in the simplest case of independent and identically distributed Bernoulli trials, the exact distributions of many run statistics still remain unknown. Departing from the traditional combinatorial approach, in this article we present a simple unified approach for the distribution theory of runs based on a finite Markov chain imbedding technique. Our results cover not only the identical Bernoulli trials, but also the nonidentical Bernoulli trials. As a byproduct, our results also yield the exact distribution of the waiting time for the mth occurrence of a specific run.

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381 citations

"Distributions of Patterns of Pair o..." refers background in this paper

...The books by Godbole and Papastavridis [8], Balakrishnan and Koutras [9], Fu and Lou [10] provide excellent information on past and current developments in this area....
[...]
...[9] N. Balakrishnan and M. V. Koutras, Runs and Scans With Applications, John Wiley and Sons, New York, NY, USA, 2002....
[...]
...References [1] J. C. Fu and M. V. Koutras, “Distribution theory of runs: a Markov chain approach,” Journal of the American Statistical Association, vol. 89, no. 427, pp. 1050–1058, 1994....
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...A considerable amount of literature treating waiting time distributions have been generated, see Fu and Koutras [1], Aki et al....
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...[3] M. V. Koutras, “Waiting time distributions associated with runs of fixed length in two-state Markov chains,” Annals of the Institute of Statistical Mathematics, vol. 49, no. 1, pp. 123–139, 1997....
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Journal Article•DOI•

On probability generating functions for waiting time distributions of compound patterns in a sequence of multistate trials

[...]

James C. Fu¹, Yung-Ming Chang•Institutions (1)

University of Manitoba¹

01 Mar 2002-Journal of Applied Probability

TL;DR: In this paper, a simple way to obtain the probability generating functions for waiting time distributions of compound patterns by using the finite Markov chain imbedding method has been provided, and a computer algorithm based on Markov Chain Imbedding technique has been developed for automatically computing the distribution, probability generating function and mean of waiting time for a compound pattern.

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Abstract: Probability generation functions of waiting time distributions of runs and patterns have been used successfully in various areas of statistics and applied probability. In this paper, we provide a simple way to obtain the probability generating functions for waiting time distributions of compound patterns by using the finite Markov chain imbedding method. We also study the characters of waiting time distributions for compound patterns. A computer algorithm based on Markov chain imbedding technique has been developed for automatically computing the distribution, probability generating function, and mean of waiting time for a compound pattern.

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56 citations

"Distributions of Patterns of Pair o..." refers methods in this paper

...Fu and Chang [12] developed general method based on the finite Markov chain imbedding technique for finding the mean and probability generating functions of waiting time distributions of compound patterns in a sequence of i....
[...]
...Fu and Chang [12] developed general method based on the finite Markov chain imbedding technique for finding the mean and probability generating functions of waiting time distributions of compound patterns in a sequence of i.i.d. or Markov dependent multistate trials....
[...]
...[12] J. C. Fu and Y. M. Chang, “On probability generating functions for waiting time distributions of compound patterns in a sequence of multistate trials,” Journal of Applied Probability, vol. 39, no. 1, pp. 70–80, 2002....
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Journal Article•DOI•

Sooner and later waiting time problems for success and failure runs in higher order Markov dependent trials

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Sigeo Aki¹, Narayanaswamy Balakrishnan², Sri Gopal Mohanty²•Institutions (2)

Osaka University¹, McMaster University²

01 Dec 1996-Annals of the Institute of Statistical Mathematics

TL;DR: In this article, the probability generating functions of the waiting times for the first success run of length k and for the sooner run and the later run between a success run and a failure run in the second order Markov dependent trials were derived using the probability generator function method and the combinatorial method.

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Abstract: The probability generating functions of the waiting times for the first success run of length k and for the sooner run and the later run between a success run of length k and a failure run of length r in the second order Markov dependent trials are derived using the probability generating function method and the combinatorial method. Further, the systems of equations of 2.m conditional probability generating functions of the waiting times in the m-th order Markov dependent trials are given. Since the systems of equations are linear with respect to the conditional probability generating functions, they can be solved exactly, and hence the probability generating functions of the waiting time distributions are obtained. If m is large, some computer algebra systems are available to solve the linear systems of equations.

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51 citations