Discussion10

pg 414

3. In Example 11.5, find P, P^2, and P^3. What is P^n?

Example 11.5 Each time a certain horse runs in a three-horse race, he has probability 1/2 of winning, 1/4 of coming in second, and 1/4 of coming in third, independent of the outcome of any previous race. We have an independent trials process, but it can also be considered from the point of view of Markov chain theory. The transition matrix is

Answer:

P <- matrix(c(0.5, 0.25, 0.25,
             0.5, 0.25, 0.25,
             0.5, 0.25, 0.25), nrow = 3, ncol = 3, byrow=T)

# for P
P

##      [,1] [,2] [,3]
## [1,]  0.5 0.25 0.25
## [2,]  0.5 0.25 0.25
## [3,]  0.5 0.25 0.25

# for P^2

P_square <- P %*% P
P_square

##      [,1] [,2] [,3]
## [1,]  0.5 0.25 0.25
## [2,]  0.5 0.25 0.25
## [3,]  0.5 0.25 0.25

# for P^3
P_cube <- P_square %*% P
P_cube

##      [,1] [,2] [,3]
## [1,]  0.5 0.25 0.25
## [2,]  0.5 0.25 0.25
## [3,]  0.5 0.25 0.25

# for P^n lets say value for n = 4, now
P_value_n <- P_cube %*% P
P_value_n

##      [,1] [,2] [,3]
## [1,]  0.5 0.25 0.25
## [2,]  0.5 0.25 0.25
## [3,]  0.5 0.25 0.25

Finally, we can say that the increment on the n value doesn’t make any changes to the P value.

So, P^n = P.