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Success run statistics defined on an urn model

Part of: Combinatorial probability Distribution theory

Published online by Cambridge University Press:  01 July 2016

Frosso S. Makri ,
Andreas N. Philippou  and
Zaharias M. Psillakis
Frosso S. Makri*
Affiliation:
University of Patras
Andreas N. Philippou*
Affiliation:
University of Patras
Zaharias M. Psillakis*
Affiliation:
University of Patras
*
∗ Postal address: Department of Mathematics, University of Patras, 26500 Patras, Greece.
∗ Postal address: Department of Mathematics, University of Patras, 26500 Patras, Greece.
∗∗∗∗ Postal address: Department of Physics, University of Patras, 26500 Patras, Greece. Email address: psillaki@physics.upatras.gr
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Abstract

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Statistics denoting the numbers of success runs of length exactly equal and at least equal to a fixed length, as well as the sum of the lengths of success runs of length greater than or equal to a specific length, are considered. They are defined on both linearly and circularly ordered binary sequences, derived according to the Pólya-Eggenberger urn model. A waiting time associated with the sum of lengths statistic in linear sequences is also examined. Exact marginal and joint probability distribution functions are obtained in terms of binomial coefficients by a simple unified combinatorial approach. Mean values are also derived in closed form. Computationally tractable formulae for conditional distributions, given the number of successes in the sequence, useful in nonparametric tests of randomness, are provided. The distribution of the length of the longest success run and the reliability of certain consecutive systems are deduced using specific probabilities of the studied statistics. Numerical examples are given to illustrate the theoretical results.

Keywords

Success runwaiting timeurn modelPólya-Eggenberger sampling schemelinear and circular binary sequencesBernoulli trialnonparametric test of randomnessreliability of consecutive systems

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 62E15: Exact distribution theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 4 , December 2007 , pp. 991 - 1019
Copyright
Copyright © Applied Probability Trust 2007 

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