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 (Xn,Λ 1,Yn,Λ2 ) using method of conditional probability generating functions where Xn,Λ1(Yn,Λ2) denotes the number of occurrences of pattern Λ1(Λ2 ) 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.
Highlights
The distributions of several run statistics are used in various areas such as reliability theory, testing of statistical hypothesis, DNA sequencing, psychology [1], start up demonstration tests [2] etc
The probability distribution of various run statistics associated with the above counting schemes have been studied extensively in the literature in different situations such as independent Bernoulli trials (BT), non-identical BT, Markov dependent BT (MBT), higher order MBT, binary sequence of order k, multi-state trials etc
Very little work is found on the distribution theory of run statistics in case of bivariate trials which has applications in different areas such as start up demonstration tests with regard to simultaneous start ups of two equipment, reliability theory of two dimensional consecutive k, r out of k 1, n : F -Lattice system etc as specified by
Summary
The distributions of several run statistics are used in various areas such as reliability theory, testing of statistical hypothesis, DNA sequencing, psychology [1], start up demonstration tests [2] etc. Even though the distribution of waiting time of the pattern of general shape in the sequence of multi-variate trials with i.i.d. components has been done, the joint distribution of number of occurrences of patterns i , i 1, 2, , m in the sequence of ith component of the m -variate trials X 1, X 2 , , X n is still unknown. We study this joint distribution of runs under the non-overlapping counting scheme of runs by using the method of conditional pgfs.
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