Runs and Patterns in a Sequence of Markov Dependent Bivariate Trials

(1)

doi:10.4236/ojs.2011.12014 Published Online July 2011 (http://www.SciRP.org/journal/ojs)

Runs and Patterns in a Sequence of Markov Dependent

Bivariate Trials

Kirtee Kiran Kamalja, Ramkrishna Lahu Shinde

Department of Statistics, School of Mathematical Sciences, North Maharashtra University, Jalgaon, India

E-mail: [email protected]

Received May 13, 2011; revised June 2, 2011; accepted June 8, 2011

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 n , 0

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

, , , ; , , ,

n k n k n k n k

X X Y Y where for i0,1, _,1 de-

i i n k

X

notes the number of occurrences of i-runs of length in the first component and _, 2

i

n k denotes the number

of occurrences of i-runs of length k in the second component of Markov dependent bivariate trials. Further we consider two patterns

1

i

k Yi

2

i

1

 and ₂ o lengths k₁ and k₂ respectively and obtain the pgf of joint

dis-tribution of



X ,₁ ,₂



g method of conditional probability generating functions where f

,

n Yn usin Xn,1

 

Yn,2

number of occurrences of pattern

denotes the  1

 

2 of length 1

 

k2 in the 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 distribu-tions are studied using the joint distribution of runs.

k first (

Keywords: Markov Dependent Bivariate Trials, Conditional Probability Generating Function, Joint Distribution

1. Introduction

The distributions of several run statistics are used in various areas such as reliability theory, testing of statis-tical hypothesis, DNA sequencing, psychology [1], start up demonstration tests [2] etc. There are various count-ing schemes of runs. Some of the most popular countcount-ing schemes of runs are non-overlapping success runs of length [3], overlapping success runs of length [4], success runs of length at least , - overlapping suc-cess runs of length [5], sucsuc-cess runs of exact length

[6].

k k

k 

k k

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

k

 

k r, out of



k1,n



: 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.

Consider a sequence



Xn,n0



 a a

of -valued trials where is set of all possible outcomes of trials under study. The simple pattern is composed of specified sequence of states i.e. _{1 2} k

S S

k   a where

1, ,2

a a akS. The number of occurrences of patterns

(2)

counting scheme of patterns allows an overlap of pre- specified fixed length in the successive occurrences of patterns.

Recently the study of distributions of different statis-tics based on patterns has become a focus area for many researchers due to its wide applicability area. Distribu-tion 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 .

, ,

r k

W 



,

th

r

r k

, X X

, ,

r k

Joint distribution of number of occurrences of pattern

1 of length k1 and pattern 2 of length 2 in n

Markov dependent multi-state trials is studied by [11]. [12] considered a sequence

W _ 1

 k

, 1, 2,

i

X i  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].

m

 

0,1

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

, 1, 2, ,

i i

  

m



m ith

1, 2, , 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 distribu-tion of number of occurrences of patterns in both com-ponents of the bivariate trials.

In this paper we consider the sequence  

of -valued Markov dependent bivariate trials. In

Sec-trials. In Section

. The Joint Distribution of Number of

Let

, 0

n

Y 

 

 

 

n

X

  

 

S

tion 2, we obtain the pgf of joint distribution of number

of occurrences of i-runs of length k_i1 in first compo-nents and i-runs of length k_i2 in t second compo-nents of the bivariate trials



i 0,1



. 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 k1 in

the first component and number of o rrences of pattern

2

 of length k2 in the second component of bivariate

4, we develop an algorithm to evaluate the exact probability distributions of the random vari-ables 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 pat-terns in bivariate trials. In Section 6 we present some numerical work based on distribution of runs and pat-terns. Finally in Section 7, we discuss an application and

generalization of the studied distributions.

he

cu



e non-o

c

2 Occurrences of Runs

0 0 1 1

, , ,

0 1 0 1

S        _{       }_

       

 .

, 0

n n

n

 Consider the sequence

X

Y

  

 _ 

  

  

  of S-valued Markov dependent bivari-

ate trials with the transition probabilities,

1

i i

X x X u u

        

, 1

, 1, 2, ,

uv xy

i i

x

P p S

Y y Y v v y

i n



  

    

_ _ _{ } _ _ _{ } _{   } _ _ _{ } _ _ _{ } _{   }

 

 

and the initial probabilities

0 0

πxy

P

Y y

       

 

X x

   x

S y

       .  

for

Let

 

 

1 2

,_i ,_i

i i

n k n k

X Y

1

be the number of -runs ( = 0,1) of le

i i

ngth ki

 

ki

mpon

2 _i_n _te

n trials associa d with the first

(second ent bivariate trials (i.e. number of i-runs in

) co of



 

0, 1, , n 0, , ,1 n

X X  X Y Y  Y . In this section

e derive the

w joint distribution of



1 1 2 2



0 _, 1 _; 0 _, 1

X X Y Y .

0 1 0 1

, , , ,

n k n k n k n k



0, ; ,1 0 1



n t t s s

 be the pgf of distribution of

Let



1 1 2 2



0 1 0 1

1 , , , ; , , ,

n k n k n k n k

X0 X1 Y0 Y . Assume that for a non-negative

integer cn, we hav

have obs c). We

fine

e observed until





th

n c trial

(i.e. we erved Xi, 0,1, ,  de-

i

i n

Y

 

  







i j







0, ; ,1 0 1

i j

c c t t s s

 

l distribution of number of i-run

1 2 1 2

, ; , , ; ,

i m j m i m j m

as pgf of

condi-tiona s of length 1

i

k in 1, ,

n c n

X    X and number of j-runs of length k2j in

ven that we have bserved

n c

X 

 

ently h

1, , n

Y Y gi o

X

n c 

0 1

, , ,

n c

X

Y Y 

        

 

   

1

Y _ and curr ave i-run of

length mi in first component and -runs of length j

2

j

m in second component of bivariate trials is

ob-ed, 1 _{1, 2, ,} 1

i i

m   k , 2 1, 2, , 2

the

serv mj   kj, i j, 0,1.

We define c 1

a  a for a0,1.

Now by assuming π00 1, we have,





 

_

_(2.1)

Also we have,



0,0;0,0

0, ;1 0 ,

t t s t t

n ,s1 n 0 1; ,s s0 1



_, 1_{; ,} 2



_

i j

i m j m

t t





1 1

0 0 1 0 1

2 2

, ; , 1 for 1, 2,..., , 1, 2,..., and , 0,1

i i

j j

s s m k

m k i j

 

  (2.2)

(3)

Conditioning on the next trial from each stage, we ha



if 2

j 



if _j 

1, 2, , c  n.





(0,0;0,0)

0 1 0 1

(0,1;0,1) (0,1;1,1) 00,00 1 00,01 1

(1,1;0,1) (1,1;1,1) 00,10 1 00,11 1

, ; ,

n

n n

t t s s

p p



 

 

 

 

ve the following system of recurrent relations of con-ditional pgfs for 1, 2, ,c  n and i j, 0,1.



_, 1_{; ,}

i i m j















2

1 2 2

1

, 1; , 1 ,1; , 1

, 1 _, 1

, 1; ,1 ,1; ,1

1 1

, ,

j

c

i j j

c

c c c

i

c c c

m c

i m j m i j m

ij ij c _{ij i j} c

i m j i j

c c

ij ij ij i j

p p



 

  

 



 

 

 

1 _{1, 2, ,} 1 ₂

i i

m   k  ,

2

2 _{1, 2, ,} ₂

j

m   k ,

















1 2 2

1

, ; ,

, 1; , 1 ,1; , 1

, 1 _, 1

, 1; ,1 ,1; ,1

1 1

, ,

i j

c

i j j

c

c c c

i

c c c

i m m c

i m j m i j m

ij ij c i _{ij i j} c

i m j i j

c i c

ij ij ij i j

p t p



 

  

 



 

 

 

1

j

1 1 ₁

i i

m k  ,



2



2 _{1, 2, ,} 2 ₂

j

m   k ,



c 2















1 2

, ; , ,1; , 1 ,1; , 1

, 1 _, 1

,1; ,1 ,1; ,1

1 1

, ,

i j j j

c

c c c

c c c

i m j m i j m i j m

c ij ij c _{ij i j} c

i j i j

c c

ij ij ij i j

p p

  

 

 

 

 

 

if 1 1

i i

m k , m2_j 1, 2, , k2_j 2





















1 2 1 2 2

1

, ; , , 1; , 1 ,1; , 1

, 1 _, 1

, 1; ,1 ,1; ,1

1 1

, ,

c

i j i j j

c

c c c

i

c c c

i m j m i m j m i j m

c ij ij c j _{ij i j} c

i m j i j

c c

ij ij ij i j

j

p s p s

p p

  

 

  

 



 

 

 

if 1 _{1, 2, ,} 1 ₂

i i

m   k  , m2j k2j1,





















1 2 2 2

1

, ; , , 1; , 1 ,1; , 1

, 1 _, 1

, 1; ,1 ,1; ,1

1 1

, ,

i j i j j

c

c c c

i

c c c

i m j m i m j m i j m

c ij ij c i j _{ij i j} c

i m j i j

c i c

ij ij ij i j

1 c

j

p t s p s

p t p

  

 

  

 



 

 

 

if 1 1 ₁

i i

m k  , m2j k2j1,

















1 2 2 2

, ; , ,1; , 1 ,1; , 1

, 1 , 1

,1; ,1 ,1; ,1

1 1

, ,

c

i j j j

c

c c c

c c c

i m j m i j m i j m

c ij ij c j _{ij i j} c

i j i j

c c

ij ij ij i j

p s p

p p

  

 

 

 

 





j

s



if 1 1

i i

m k , m2_j k2_j1,















1 2 1

1

, ; , , 1; ; ,1

, 1 _, 1

, 1; ,1 ,1; ,1

1 1

, ,

i j i

c

c c c

i

c c c

i m j m i m j j

c ij ij c ij i j c

i m j i j

c c

ij ij ij i j

p p

  

 



 



 

 

 





,1 _ic,1

if 1 _{1, 2, ,} 1 ₂

i i

m   k  , m2_j k2_j,





















1 2 1

1

, ; , , 1; ,1 ,1

, 1 _, 1

, 1; ,1 ,1; ,1

1 1

, ,

c

i j i

c

c c c

i

c c c

i m j m j i

c ij ij c i _{ij i j} c

i m j i j

c i c

ij ij ij i j

p t p

  

 



 



 

 

 

; ,1

i m j

1 1 ₁

i i

m k  , m2_j k2_j if





 













2

, ; , _{,1; ,1} ,1; ,1

, 1 _, 1

,1; ,1 ,1; ,1

1 1

, ,

c j

c

c c c

c c c

i m j m i j i j

c ij ij c _{ij i j} c

i j i j

c c

ij ij ij i j

p p

  

 

 

 

 

1

i

1 1

i i

m k , m2_j k2_j if

Thus we have



1 1



2 2



t

0 1 0 1 k k k k

for c 1, 2, ,

recurren relations of conditional pgfs   n1 and these can be

written as,









0 1

1 1 1 1

1 2 12

0 1 0 1 1

0 0 0 0

,

, ; ,

i i j j ij i j _c

i j i j

s 0, ;1

c t t s

A B t B s B t s  _ t t s s

   



 

  

 









(2.3) where







2 0

2 2 1 2

1 1 1 1

(0,1;0, )

(0,0;0,0) (0,1;0,1) (0,1;1,1)

(0,1;1, ) (1,1;1, ) (1, ;1, )

k

c c c c

c

k k k k

c c c

     

   

 

and 1 1 1 2 1 1 12

0 i i 0 j j 0 0 ij i j

i j i j

A B t B s B t s

   













is a square x of size

matri



k₀k₁



k₀ k₁



. to corresp

1 1 2 2 _Each_{row of this m}

a-trix corresponds onding element of



0, ; ,1 0 1



c t t s s

 and its elements are the coefficients of elements of



₀ ₁ ₀ ₁



1 , ; ,

c t t s s .

For c 0



 , we have,



0 1



0 t ,t s s1; ,0 1

 

where 1 is the column vector with all the elements equal to 1.

Using (2.3) recurrently for c1, 2, , n, we have,



0, ; ,1 0 1



n t t s s





1 1 1 1

1 2 12

0 1 0 1 0

0 0 0 0

, ; ,

n

i i j j ij i j

i j i j

A B t B s B t s t t s s



   

 

_    _









(2.4) Hence from (2.1) and (2.4) we get the pgf of dis

tio

tribu-n of



1 1 2 2



0 1 0 1

, , , ; , , ,

n k n k n k n k

X0 X1 Y0 Y1 as,



0 1 0 1



1 1 1 1

1 2 12

0 0 0 0

, ; ,

t t s s

1

n

i i j j ij i j

i j i j

p A B t B s B t s

   

 



 _    _









(2.5)

(4)

where p





is the first column of identity matrix of order



3. The Joint Distribution of Number of

Let and be two

pat-nsider of patterns in

of pattern



1 1 2 2

0 1 0 1 k k k k .

Occurrences of Patterns

1

1 a a1 2 ak

  

ch to count the num

2

2 b b1 2 bk

  

heme er of occurrences terns of lengths k1 and k2 respectively where

 

, 0,1

i j

a b  , i1, 2, , ; k j₁ 1, 2, , k₂. We co non-overlapping counting sc

the hi

w b 

urs.

in a e of has to restart

counting from scratch each time the pattern  1

,

n no verlapping

oc-currences of patterns 1 in the first component of n

bivariate trials (i.e. in 1, 2, , n

given sequenc X de tes th

n trials, one

mber of non-o occ

Let e nu

X X  X ) and

2

,

n

Y _ d notes the number of non-ov lapping occurrences of pat-terns 2 in the second component of n bi riate

tri-als ( Y Y1, , ,2 Yn). Consider the random vector



Xn,1,Yn,2



. In this section we obtain the pgf of joi

distribution of



Xn_,₁,Yn_,₂



ethod of ndi-tional pgfs.

Let of joint distribution of Xn_,₁,Yn_,₂ be





e-nt er

using m va

i.e. in

co

pgf





 

1, 2

n t t t

 n . Assum

, we have observed until

. For cn, we define, the following condition

Let

e that for a non-negative integer



th

c

 trial i.e.

1 _, 2 _,

X X

Y Y

 

   

   

cn



n

1  2  Yn c 

al pgfs.

, Xn c

 



 

( , )i j

c t

 be pgf of condition ribution of

r terns  , ,

al dist

number of occu rences of pat 1 in Xn c 1 Xn

and number of occurrences of pattern 2 in 1

n c  n

at





th

n c trial) we ha , ,

Y  Y of bivariate trials gi ve obser no

ven that currently (i.e.

ne of the sub- ved

patterns of ₁ and ₂ and n c n c

X i

Y j



   _

  

 

  , where

i S

     .

j

 

Similarly let,  ,₁

 

,

i j c

 _ be pg nal distri

tion of num er of

t f of conditio

bu-b occurrences of patterns 1 in 1, ,

n c n

X _{ }  X bivariate trials give served the su co

and patterns in of

n that at

b-pattern of first

one of

2





n c



h i

1

n c  

trial we

 in

, , n

Y Y

have ob-the

th

lengt of 1

mponent and n the sub pattern of 2 is

ob-served in the second component of bivariate trials and

n c i

n c

Y j



   _

X   a

 

  where i1, 2 k and j0,1.

Let  

1

, ,

 

2

, ,

i j c t

 _ be pgf of conditional distri tion of

number of occurrences of patterns ₁ in n c₁, , n

bu X _{ }  X and patterns ₂ in Y Y of bivariate trials





th

n c e obse

1, ,

n c   trial we h in the fi

n

av rst com given that at

the sub-pattern of pon

pattern of the

e ob d

rved none of ent and a sub

1



length j of ₂ in the second component of bivariat trials _{is serve} and n c

j

n c b

Y





  

 

   

where i 0,1

i

X   

 and j1, 2, , k2.

Also let  ,₁ ₂

 

, ,

i j

c t

   as pgf of onditional distribution of c

pattern

number of occurrences of 1 in Xn c 1, , Xn

and pattern 2 in Yn c 1, ,Yn of bivari

of leng f 1

ate trials given he sub-pattern that at





th obse ved t

n c

o



th i

trial we have r

 in the co

and

first mponent and a sub pa

iat s

ttern of the length j of ₂ in the second component of bivar e trial n c i

j n c

a X

b Y



 

   

 



  where

1

1, 2, ,

i  k and j1, 2, , k₂. Let,

1

c

a_i  a_i, i1, 2, , k₁ and c 1

j j

b  b , j1, 2,,k₂. For cn, we assume

i

1

c i

that at

1



If





th

n c observed i

ondition

trial, the

su of s n the first

compone riate trials. we c on the next trial as

b-pattern of nt of biva

n c n c

X

Y_

 

length i

1 a

   

 

1 yn c1

  

 

  

   i.e. of leng-

th i of ₁

 the sub-pattern

 observ d at



th

n c trial breaks at



1



e



st

n c  trial then to check whether any sub-pattern of ₁ of length r r



i



1

has occurred, we define the indicator function ir as,



1ai r 2,a2ai r 3,...,ar1a ai, raic1



1

, 1;

i k r i

1

ir I a  

1, 2,

   .

Similarly we hav ndicato 

e i r function 2

jr  as,





2

1 2, 2 3..., 1 ,

c

jr I b bj r b br b bj r bj

   _{ }  _   _

1, ,

j k r j

,

j r 

2 1; b

2,

1

    .

Let





1 1 ₁

ir

r

1 max₁ , 2 m , 1, 2, ,

1, 2, 1

i i

ir r i

u  u  r i

 

   

 



 ₁ ,

ax ,

i k





2 2

2

ax and ax 1, 1, 2, ,

1, 2 , 1

j

jr jr

r j

v r r

j k

    

 



1 1

m

j

v

 

 m

,

j

Assuming π001, we 1 2 t t

have,

(3.1) We also have,





 0,0

_

_

1 2

, ,

n n t t

 

 ,

_{ }

0 1 for ;

i j i

t S

j

(5)

 

_{ }

 

_{ }

2 , 0, 1 i j t  _  1 , 1 0, 2

1 for 1, 2, , a

for 0,1 and 1, 2, , ;

i j

t i k

i j k

 _  

 



  

nd j0,1;

 

1 2

,

1 2

0, , 1 for 1, 2, , and 1, ,

i j

t i k j k

 _{ }      . (3.2)

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, 1 c n (c

t trial

 

_{ }

 

_{ }

 

_{ }

 

_{ }

_{   }

1 2 ₁ 2

1 1

1 ₁

1

1 1 1 1

, , 1,

, 1, 1 1, , , c c c c

c c c

c ij _{ij a b}

a b b

c c

ij a b ij a b

t p t p t

p  t p  t

        1 1 1 ,1 1,1 , , 1, c a i j

a b c c

  _{ }  

if i j, 0,1,  

_{ }

 

_{ }

 

_{ }

 

_{ }

_

_





_{ }





_{ }

_

_





_{ }

1 1

1 1 2

2 1

1 2 2

1 1

2 1 1 1

1 1 1 1 1 1 1 , 1,1 ,

, 1, ,

,1 ,1

1 1

1, , 1,

,

, ,

1 1 1

1, , 1, 1, , 1 i i i c i c i _i

i c c c

i c c i _i c c i i

i j i

a j a b

c c

u _i a _i

c c

a j a b

u b _i a b _i

c c

a j a b i b c a j a b

t p t

p t u t u

p t                                  _   _      _  _    if 1 1

1, 2, , 2; ,1 i  k  j0  

_{ }

 

_{ }

 

_{ }

 

_{ }

_

_

 





 









_{ }

1 1

1 1 2

2 1 2 2 1 1 1 1 2 1 2 1 1 1 1 1 1 , 1,1 , 1

, 1, ,

,1 ,1 1 1 1, 1, , , ( , )

1 1 1

1, , 1, 1 1, , 1 1 i i i c i c i _i c c

i c _i

c c i _i

c c i i

i j i

a j a b

c c

u _i a

c c

a j a b

a b

u b i i

c c

a j a b i b c a j a b

t p t t

p t u t

p t t

                              i u u  _   _      _   _    if 

1 1; 0,

i k j 1

 

_{ }

 

_{ }

 

_{ }

 

_{ }

_{   }

1 1 1

1 1 2 ₁

1 1

1 ₁

1

1 1 1 1

,1

, 1,1

,

, 1, , _,

, 1, 1 1, , , c c i i c c c

c c c

i i

a i j

a j a b

c c _{a j a b} c

a b b

c c

a j a b a j a b

t p t p t

p t p

                 2 1, t 

if 1

Similarly we have recurrent relations of the condi-tional pgfs

1, 0,

ik j .

 

_{ }

2 , , i j c t

 _ for k as

follows. 2

0,1 and 1, 2, ,

i j 

 

_{ }

 

_{ }

   





_{ }

_

_

   





_{ }

_

_





_{ }

1 1

2 1 2

2 1

1 1

1 2 1 1

2 1 1

1 2 1 1

, 1, 1

,

, 1, ,

,

, ,

1 1 1

1, , , 1 1, , 1 j j c j j j

c c c

j c c j j c c j j

i j j

ib a b

c c

a b

a v _j a b _j

c c

ib a b a j c ib a b

t p t

p t v t v

p t                              _   _    if 1 2 1 1, 1, 1 1

1, , 1, 1

j c

j

v _j b _j

c c

ib

p _{  } t v _{ } t v 

  

2

1, 2, , 2;

j  k 

 

_{ }

 

_{ }

   





_{ }

_

_

   





_{ }

_

_





_{ }

1 1

2 1 2

1 2

1 2 1

1 1

1 2 1 1

2 1 1

1 2 1 1

, 1, 1

, 2

, 1, ,

1, 1,

1 1

1, , 1,

,

, ,

1 1 1

1, , , 1 2 1, , 1 1 j j j c j c j j j

c c c

j c c j j c c j j

i j j

ib a b

c c

v _j b _j

c c

ib a b

a v _j a b _j

c c

ib a b a j c ib a b

t p t t

p t v t

p t t

                                 _  _      _  _    v v  

if jk21;

 

_{ }

 

_{ }

 

_{ }

 

_{ }





_{ }

1 1 1

2 1 2 ₁

1

1 1 1

1

1 1 1 1

,1

, 1,1

,

, 1, , ,

1, , 1 1, , , c c j _j

c c c

c c c

j j

a i j

ib a b

c c ib a b c

b a b

c c

ib a b ib a b

t p t p t

p t p

                 2 1, t 

if jk2

The recurrent relations of the conditional pgfs

 

_{ }

1 2 , , , i j c t

   for i1, 2, , k1 and are as

follows. 2

1, 2, ,

j  k

 

_{ }

 

_{ }



_{  }





_{ }

_

_





_{ }





_{ }

_

_

1 1

1 2 1 2

1 2

1 2 1

1 1

2 1

1 2 1

1 1

1 1 1,

c c

i j j c

a b b

p 

   



 

, 1, 1

,

, , 1, ,

1, 1,

1 1

1, , 1,

,

, 1 , 1

1 1

1, , 1,

,

1

i j i j

j c

i j i j

i c

i j i j

i

i j i j

a b a b

c c

i v _j i b _j

c c

a b a b

u j _i a j _i

c c

a b a b a

t p t

p t v t v

p t u t u

                                       _   _      _   _  

 





_{ }

_

_



_{  }

_

_





_{ }

_

_

_

1 2 2 1 1 1 2 2 1 1 , 1 1 1, , 1 1 1, ,

1 1 1

1 1 1 1 i c j j c i c c i j

u b _i _j

c

a v _i _j

c

a b _i _j

c

t u v

t u v

t u v

                    _  if

2 2,

1 1 ,

j i

u v _i _j

t u v



  

1 2

1, 2, , 2; 1, 2, , 2

i  k  j  k 

 

_{ }

 

_{ }



_{  }





_{ }

_

_





_{ }





_{ }

_

_

1 1

1 2 1 2

1 2

1 2 1

1 1

2 1

1 2 1

1 1

, 1, 1

, 1

, , 1, ,

1, 1,

1 1

1, , 1,

,

, 1 , 1

1 1

1, , 1,

,

1

i j i j

j c

i j i j

i c

i j i j

c c i j i j

i j i j

a b a b

c c

i v _j i b _j

c c

a b a b

u j _i a j _i

c c

a b a b c a b a b

t p t t

p t v t

p t u t

p                                           _  _      _  _   

 

1 v t u  

 





_{ }

_

_



_{  }

_

_





_{ }

_

_

_

2 2 1 2 2 1 1 1 2 2 1 1 , 1 1 1, , , 1 1 1, , 1 1 1, ,

1 1 1

1 1 1 1 j i i c j j c i c c i j

u v _i _j

u b _i _j

c

a v _i _j

c

a b _i _j

c

t u v

t u v

t u v

t u v

                          _ 

(6)

 

_{ }

 

_{ }

   





_{ }

_

_





_{ }

   





_{ }

_

_

1 1

1 2 1 2

1 2

1 2 1

1 1

1 2 1 1

2 1 1

, 1, 1

,

, , 1, ,

1, 1,

1 1

1, , 1,

,

, 1 1, ,

, ,

1 1 1

1, ,

1

i j j

j c

i j j c c i j j

j

c c c

j c c

i j j

a b a b

c c

v _j b _j

c c

a b a b a j c a b a b

a v _j a b _j

c c

a b a b

t p t

p t v t

p t

p t v t

                                  _  _     v v    _   _   1 2

if ik j; 1, 2, , k 2  

_{ }

 

_{ }



_{  }





_{ }

_

_





_{ }





_{ }

_

_

1 1

1 2 1 2

1 2

1 2 1

1 1

2 1

1 2 1

1 1

, 1, 1

, 2

, , 1, ,

1, 1,

1 1

1, , 1,

,

, 1 , 1

1

i

u ₁ ₂

1, , 1,

,

1

i j i j

j c

i j i j

i c

i j i j c c i j i j

i j i j

a b a b

c c

i v _j i b _j

c c

a b a b

u j a j _i

c c

a b a b c a b a b

t p t t

p t v t v

p t t u t

p                                           _   _      _   _   

   

 







  





 







2 2 1 2 1 2 2 1 1 2 1 1 , 1 1 1, , , ( , )

1 1 1 1

1, 1,

( , )

1 1 1

1 1 1 1 j i j c

i c _i

j c c i j

u v _i _j

a v

u b i j i j

c c

a b i j

c

t u v

t u v t u v

t u v

                          1 2

if i1, 2, , k 2;jk 1  

_{ }

 

_{ }



_{  }





_{ }

_

_





_{ }





_{ }

_

_

1 1

1 2 1 2

1 2

1 2 1

1 1

2 1

1 2 1

1 1

1

, 1, 1

, 1 2

, , 1, ,

1, 1,

1 1 t1

1, , 1,

,

, 1 , 1

1 1

1, , 1,

,

1

i j i j

j c

i j i j

i c

i j i j c i j i j

i j i j

a b a b

c c

i v _j i b _j

c c

a b a b

u j _i a j _i

c c

a b a b a b a b

t p t t t

p t v t v

p t u t u

p                                          _   _      _   _   

 

2 t

 





_{ }

_

_



_{  }

_

_





_{ }

_

_

_

2 2 1 2 1 1

2 1 2

1 2 1 1 , 1 1 1, , , ,

1 1 1 1

1, 1,

,

1 1 1

1 1 1 1 j i c j c i c i j c c i j

u v _i _j

c

a v

u b _i _j _i _j

c c

a b _i _j

c

t u v

t u v t u v

t u v

                           _  1 2

if i k 1;jk 1  

_{ }

 

_{ }

   





_{ }

_

_





_{ }

   





_{ }

_

_

1

1 2 1 2

1 2

1 2 1

1 1

1 2 1 1

2 1 1

, 1, 1

, 2

, , 1, ,

1, 1,

1 1

1, , 1,

, , 1 2 1, , , ,

1 1 1

1, ,

1

i j i j

j c

i j j c c i j j

j

c c c

j c c

i j j

a b a b

c c

v _j b _j

c c

a b a b a j c a b a b

a v _j a b _j

c c

a b a b

t p t t

p t v t

p t t

p t v t

                                  _  _     v v    _   _   1 2

if ik j; k 1

 

_{ }

 

_{ }

 

_{ }

 

_{ }

_

_





_{ }





_{ }





_{ }

_

_

1

1 2 1 2

2 1

1 2 2

1 1

1 1 1

2 1 1 1

1 1 1

, 1,1

,

, , 1, ,

,1 ,1

1 1

1, , 1,

,

1, 1, ,

, ,

1 1 1

1, ,

1

i j i i

i c

i j i

c i c i j i

i c c c

i c c

i j i

a b a b

c c

u _i a _i

c c

a b a b i b c a b a b

u b _i a b _i

c c

a b a b

t p t

p t u t

p t

p t u t

                                  _  _     u u    _   _   1 2

if i1, 2, , k 2;jk  

_{ }

 

_{ }

 

_{ }

 

_{ }

_

_





_{ }





_{ }





_{ }

_

_

1

1 2 1 2

2 1

1 2 2

1 1

1 1 1

2 1 1 1

1 1 1

, 1,1

, 1

, , 1, ,

,1 ,1

1 1

1, , 1,

, 1, 1 1, , , ,

1 1 1

1, ,

1

i j i i

i c

i j i

c i c i j i

i c c c

i c c

i j i

a b a b

c c

u _i a _i

c c

a b a b i b c a b a b

u b _i a b _i

c c

a b a b

t p t t

p t u t

p t t

p t u t

                                  _  _     u u    _   _   1 2

if i k 1;jk  

_{ }

 

_{ }

 

_{ }

 

_{ }





_{ }

1 1 1

1 2 1 2 _{1 1} 2

1 1 1

1

1 1 1 1

,1

, 1,1

,

, , 1, , _, 1,

1, , 1 1, , , c c

i j _{i j}

c c c

c c c

i j i j

a i j

a b a b

c c _{a b a b}

b a b

c c

a b a b a b a b

t p t p t

p t p t

                   c 1 2

if ik j; k The above system of



k1 2

 

k22



fs  

recurrent re-lations of conditional pg ci j,

 

t ,



i j, 0,1



;

 

_{ }

1 , , i j c t

 _ ,



i0,1, , ; k₁ j0,1



;  i j,

 

t ,

2

,

c





i0,1;j1, 2, , k2



and

 

_{ }

1 2 , , , i j c t  _{ }



i1, 2, , ; k j1 1, 2, , k2



can be written as follows.

 





 





1 1 2 2 12 1 2 ₁

1 1 2 2 12 1 2

if 2,3, ,

1 if 1

c c

A B t B t B t t t c n

t

A B t B t B t t c

  _            (3.3) where 1 is column vector with all its elements 1 and

 

n t

 is column vector with its elements as follows.              



       



1

1 1 2

2 1 2

2 1 2 1 2 1 2

1,0 ,1 0,1 0,0 0,1 1,0 1,1

, , ,

1, 1,1 1,2 ,

, , , , , , ,

... k

c c c c c c c

k k k

c c c c

                      

From (3.1) we get,



1,2



'



1,2



n t t p _n t t

  

p is first column of identity matrix of order

where



k1 2

 

k22



.

The recurrent use of (3.3) gives the pgf of



Xn,₁,Yn,₂



as follows.

 

'



1 1 2 2 12 1 2



1

n

n t p A B t B t B t t