Introduction to Probability, Grinstead, C. Snell, J.,
1997
Page 443
Question 5
Question
Find the fixed probability vector for each of the following
regular matrices
\(\begin{pmatrix}0.75 & 0.25 \\0.5 &
0.5 \\\end{pmatrix}\)
#install.packages("markovchain")
library(markovchain)
matrix <- matrix(c(.75, .25, .5, .5), byrow = TRUE, nrow = 2)
chain <- new("markovchain", states = c("1", "2"), transitionMatrix = matrix)
answer <- steadyStates(chain)
answer## 1 2
## [1,] 0.6666667 0.3333333\(\begin{pmatrix}0.9 & 0.1 \\0.1 & 0.9 \\\end{pmatrix}\)
#install.packages("markovchain")
library(markovchain)
matrix <- matrix(c(.9, .1, .9, .1), byrow = TRUE, nrow = 2)
chain <- new("markovchain", states = c("1", "2"), transitionMatrix = matrix)
answer <- steadyStates(chain)
answer## 1 2
## [1,] 0.9 0.1\(\begin{pmatrix}3/4 & 1/4 & 0\\0 & 2/3 &1/3\\1/4&1/4&1/2\\\end{pmatrix}\)
#install.packages("markovchain")
library(markovchain)
matrix <- matrix(c(3/4, 1/4, 0 ,
0 , 2/3, 1/3,
1/4, 1/4, 1/2 ), byrow = TRUE, nrow = 3)
chain <- new("markovchain", states = c("1", "2","3"), transitionMatrix = matrix)
answer <- steadyStates(chain)
answer## 1 2 3
## [1,] 0.2857143 0.4285714 0.2857143