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Joint distributions of successes, failures and patterns in enumeration problems

Part of: Special processes Markov processes Distribution theory - Probability Distribution theory

Published online by Cambridge University Press:  01 July 2016

S Chadjiconstantinidis ,
D. L. Antzoulakos  and
M. V. Koutras
S Chadjiconstantinidis*
Affiliation:
University of Piraeus
D. L. Antzoulakos*
Affiliation:
University of Piraeus
M. V. Koutras*
Affiliation:
University of Athens
*
∗ Postal address: Department of Statistics and Actuarial Science, University of Piraeus, Piraeus 18534, Greece.
∗ Postal address: Department of Statistics and Actuarial Science, University of Piraeus, Piraeus 18534, Greece.
∗ Postal address: Department of Statistics and Actuarial Science, University of Piraeus, Piraeus 18534, Greece.
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Abstract

Let ε be a (single or composite) pattern defined over a sequence of Bernoulli trials. This article presents a unified approach for the study of the joint distribution of the number Sn of successes (and Fn of failures) and the number Xn of occurrences of ε in a fixed number of trials as well as the joint distribution of the waiting time Tr till the rth occurrence of the pattern and the number STr of successes (and FTr of failures) observed at that time. General formulae are developed for the joint probability mass functions and generating functions of (Xn,Sn), (Tr,STr) (and (Xn,Sn,Fn),(Tr,STr,FTr)) when Xn belongs to the family of Markov chain imbeddable variables of binomial type. Specializing to certain success runs, scans and pattern problems several well-known results are delivered as special cases of the general theory along with some new results that have not appeared in the statistical literature before.

Keywords

Markov chainswaiting timessooner waiting time problemspatternsscanssuccess runs

MSC classification

Primary: 60E05: Distributions
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K30: Applications (congestion, allocation, storage, traffic, etc.) 62E15: Exact distribution theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 3 , September 2000 , pp. 866 - 884
Copyright
Copyright © Applied Probability Trust 2000 

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