Runs and Patterns in a Sequence of Markov Dependent Bivariate Trials

n X n Y                  of 0 0 1 1 , , , 0 1 0 1 S                                    valued Markov dependent bivariate trials. By using the method of conditional probability generating functions (pgfs), we derive the pgf of joint distribution of   1 1 2 2 0 1 0 1 0 1 0 1 , , , , , ; , n k n k n k n k X X Y Y where for , de0,1 i  1 , i i n k X

valued Markov dependent bivariate trials.By using the method of conditional probability generating functions (pgfs), we derive the pgf of joint distribution of   , where for , de- notes the number of occurrences of i-runs of length in the first component and

Introduction
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.There are various counting schemes of runs.Some of the most popular counting schemes of runs are non-overlapping success runs of length [3], overlapping success runs of length [4], success runs of length at least , -overlapping success runs of length [5], success runs of exact length [6].
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 , multi-state trials etc.But 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 : F -Lattice system etc as specified by [7].[7] have studied the distribution of sooner and later waiting time problems for runs in Markov dependent bivariate trials by giving system of linear equations of the conditional pgfs of the waiting times.The distribution of number of occurrences of runs in the two components of bivariate sequence of trials and their joint distributions are still unknown to the literature.  .The number of occurrences of patterns can be counted according to the non-overlapping or overlapping counting scheme.The non-overlapping counting scheme starts recounting of the pattern immediately after the occurrence of the pattern while the overlapping counting scheme of patterns allows an overlap of prespecified fixed length in the successive occurrences of patterns.
Recently the study of distributions of different statistics based on patterns has become a focus area for many researchers due to its wide applicability area.Distribution of , the waiting time for the occurrence of pattern of length in the sequence of multistate trials is studied by [1,8,9].[10] considered the sequence 1 2 generated by Polya's urn scheme and study the waiting time distribution of for .
, , Joint distribution of number of occurrences of pattern 1 of length 1 k and pattern 2  of length 2 in n Markov dependent multi-state trials is studied by [11].[12] considered a sequence  of dimensional i.i.d.Random column vectors whose entries are -valued i.i.d.random variables and obtain the waiting time distribution of two dimensional patterns with general shape.The general method, which is an extension of method of conditional pgfs, is used to study these distributions by [12].
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 in the sequence of component of the -variate trials , , , n X X X  is still unknown.Here we derive the pgf of joint distribution of number of occurrences of runs in both the components of the bivariate trials and generalize this study to the distribution of number of occurrences of patterns in both components of the bivariate trials.
In this paper we consider the sequence   of -valued Markov dependent bivariate trials.In Sectrials.In Section

. The Joint Distribution of Number of
S tion 2, we obtain the pgf of joint distribution of number of occurrences of i -runs of length 1 i k in first components and i -runs of length .We study this joint distribution of runs under th verlapping counting scheme of runs by using the method of conditional pgfs.Further in section 3, we study the joint distribution of number of occurrences of pattern 1  of length 1 k in the first component and number of o rrences of pattern 2  of length 2 k in the second component of bivariate 4, we develop an algorithm to evaluate the exact probability distributions of the random variables under study.As an application of the derived joint distributions, in Section 5, we obtain distributions of several waiting times associated with the runs and patterns in bivariate trials.In Section 6 we present some numerical work based on distribution of runs and patterns.Finally in Section 7, we discuss an application and generalization of the studied distributions.

Occurrences of Patterns
, , n Assum , we have observed until .For c n  , we define, the following condition Let e that for a non-negative integer al dist number of occu rences of pat in of bivariate trials gi ve obser no ven that currently (i.e.ne of the sub-ved patterns of 1  and 2  and of 1 mponent and n the sub pattern of 2  is observed in the second component of bivariate trials and is serve For c n  , we as e sum has occurred, we define the indicator fu ion ir nct  as,   Similarly we hav ndicato We also have, for 1, 2, , a for 0,1 and 1,2, , ; 1 for 1, 2, , and 1, , Now for each denotes the number of trials remaining to observe) we condition on the nex to obtain the recurrent relations of conditional pgfs as follows. 2, , , The recurrent relations of the conditional pgfs 1 1 1 1 The above system of     3) where 1 is column vector with all its elements 1 and   n t  is column vector with its elements as follows.,  ; , We note that the pgf of art pg , is a p icular case of f of .Hence we develop an algorithm to the exact probability distribution of beapp ility distribution of rve that t involves matrix polynomial in , ; lied to obtain the exact probab   . Here i e is the th i row of the identity matrix of order 2.
Proof Obviously for 1 n  , we have, here O is the null m same order as that of A.
x y i j

B t s C x y A C x e y B I x e C x y e B I y e
C e e y e t s Copyright © 2011 SciRes.

OJS
Hence we get the required proof of the lemma.Theorem 4.1 The exact probability distribution o is given by, f   ' , 1 , On simplifying this expression, we have,

Waiting Time Distributions Related to
The exact probability di from its pgf given in (2.5) can be expressed as,

Runs and Patterns
, ; x y e B I y e ; 1) The components of the above expressio an be interpreted as follows.
,( , ) 3) The above interpretation is useful for deriving different waiting time distributions.Let : The waiting time for sooner occurring event bet th event ( e. j -run of leng 12 ij W events : The waiting time until the occurrence of both and , 0,1 k j  respectively.In the next sub-section we obtain the sooner waiting time distribution.

Sooner Waiting Time Distribution
The probability that 0-run of length 1 0 k (i.e.event 1 0 F ) occurs for the first time at th m trial given that none of the events 2 0 . i e P W m  written as follo s.

 
,0 S can be . Similarly the probability that 1-run of length (i.e. 0 ,0 1 ,0,0,1 ; , ; , 0,0,0,0 ' 0,0,0,0 1 ' 1 Hence from (5.4), the exact probability distrib tion of u random variable S W is given by, The pgf of is given by, Here we note that it is quiet difficult to study the later waiting time distribution between , In particular pgf of marginal distribution of , Then pgf of where O is the null matrix of order same a trix 1 0 A and 1 0 D .The pro 5.8) can be wri as follows.
bability in ( tten The above components of can be interpreted as follows.
  Using the pgf of and interpretations in .9), Particularly when 0 1 r  , we have, The pgf of is given by, Similarly waiting time distributions of : F-Lattice system can now be obtained simply by using the joint distribution of as, P(consecutive -o , , , n X X X  problem of studying distribution of reduces to the d wa e distributions can also be studied The relate iting tim fo cess as in Se e ca Markov dependent m ltivariate trials.

References
. quen . 78 llowing the same pro ction 5 in th u se of

1  and 2  o lengths 1 k and 2 k
of occurrences of i-runs of length k in the second component of Markov dependent bivariate trials.Further we consider respectively and obtain the pgf of joint dis- all possible outcomes of trials under study.The simple pattern is composed of specified sequence of states i.e.

ent and a sub 1  length j of 2 
in the second component of bivariat trials and n c

1 F , 2 1F
. To obtain the distri-bution of sooner waiting time we define the following random variables.

1 F
) occurs for the first time at tri that none of the events

x in 2 mS
can be treated as a m -digit binary number.The function   g x converts this m -digit binary number into a unique equivalent decimal number in 10 m S .Now corresponding to the m -variate sequence of 2 m S -valued Markov dependent trials  , 0 Now the original problem of studying the joint distribu- k , while[11] obtain the joint distribution of num er of occurrences of p erns in the sequence of the method of conditional pgfs.

The Joint Distribution of Number of
y satisfies (4.2). ,