Week 10 Discussion Post

Chapter 11 Exercise 2 Page 414

2. In Example 11.4, let \(a = 0\) and \(b = 1/2\). Find \(P\), \(P_2\), and \(P_3\). What would \(P_n\) be? What happens to \(P_n\) as n tends to infinity? Interpret this result.

Ans. let a = 0 and b = 1/2

P

P <- matrix(c(1,0,1/2,1/2), nrow = 2, byrow = TRUE)

P

##      [,1] [,2]
## [1,]  1.0  0.0
## [2,]  0.5  0.5

##or
fractions(P)

##      [,1] [,2]
## [1,]   1    0 
## [2,] 1/2  1/2

P2

P_2 <- P%*%P

P_2

##      [,1] [,2]
## [1,] 1.00 0.00
## [2,] 0.75 0.25

##OR
fractions(P_2)

##      [,1] [,2]
## [1,]   1    0 
## [2,] 3/4  1/4

P3

P_3 <- P%*%P%*%P

P_3

##       [,1]  [,2]
## [1,] 1.000 0.000
## [2,] 0.875 0.125

##or
fractions(P_3)

##      [,1] [,2]
## [1,]   1    0 
## [2,] 7/8  1/8

Pn

Pn <-matrix(c(1,0,1,0), byrow=TRUE, nrow=2)

Pn

##      [,1] [,2]
## [1,]    1    0
## [2,]    1    0