Page 422 Exercise 6

In the Land of Oz example (Example 11.1), change the transition matrix by making R an absorbing state. This gives

                              R         N         S
                          R   1         0         0
                          N   1/2       0         1/2
                          S   S         1/4       1/2

Find the fundamental matrix N, and also Nc and NR. Interpret the results.

P = matrix( 
    c(1,0,0,
      0.5,0,0.5,
      0.25,0.25,0.5
      ), # the data elements 
   nrow=3,              # number of rows 
   byrow = TRUE)        # fill matrix by rows 
dimnames(P) = list( 
   c("R","N", "S"), # row names 
   c("R","N", "S")) # column names 
P
##      R    N   S
## R 1.00 0.00 0.0
## N 0.50 0.00 0.5
## S 0.25 0.25 0.5
#N = inverse (I -Q)
I=diag(2)
Q = matrix(c(0,0.5,0.25,0.5),nrow=2,byrow = TRUE)
A=I-Q
N = solve(A)
N #N
##           [,1]     [,2]
## [1,] 1.3333333 1.333333
## [2,] 0.6666667 2.666667
#Nc
C=matrix(c(1,1),nrow=2,byrow = TRUE)
N%*%C
##          [,1]
## [1,] 2.666667
## [2,] 3.333333
#NR
R = matrix( 
    c(1.0,0,
      0,1
      ), # the data elements 
   nrow=2,              # number of rows 
   byrow = TRUE)
N%*%R
##           [,1]     [,2]
## [1,] 1.3333333 1.333333
## [2,] 0.6666667 2.666667