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Runs and Patterns in a Sequence of Markov Dependent Bivariate Trials

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DOI: 10.4236/ojs.2011.12014    3,255 Downloads   6,453 Views   Citations

ABSTRACT

In this paper we consider a sequence of Markov dependent bivariate trials whose each component results in an outcome success (0) and failure (1) i.e. we have a sequence {(Xn/Yn), n>=0} of S={(0/0),(0/1),(1/0),(1/1)}-valued Markov dependent bivariate trials. By using the method of conditional probability generating functions (pgfs), we derive the pgf of joint distribution of (X0n,k10,X1n,k11;Y0n,k20,Y1n,k21) where for i=0,1,Xin,k1i denotes the number of occurrences of i-runs of length k1i in the first component and Yin,k2i denotes the number of occurrences of i-runs of length k2i in the second component of Markov dependent bivariate trials. Further we consider two patterns Λ1 and Λ2 of lengths k1 and k2 respectively and obtain the pgf of joint distribution of (Xn1,Yn2 ) using method of conditional probability generating functions where Xn1(Yn2) denotes the number of occurrences of pattern Λ12 ) of length k1 (k2) in the first (second) n components of bivariate trials. An algorithm is developed to evaluate the exact probability distributions of the vector random variables from their derived probability generating functions. Further some waiting time distributions are studied using the joint distribution of runs.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

K. Kamalja and R. Shinde, "Runs and Patterns in a Sequence of Markov Dependent Bivariate Trials," Open Journal of Statistics, Vol. 1 No. 2, 2011, pp. 115-127. doi: 10.4236/ojs.2011.12014.

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