Question

(8 points) The transition matrix P for a Markov chain is shown below. Find the stable vector W for this Markov chain. Write the entries as whole numbers or fractions in lowest terms T0 0 0 1 P- 1 프 00 1 3 Wー

Answer 1

Let W = [a, b, c, d] where a + b + c + d = 1

Then WP = W which gives below set of equations.

(1/4)c + (1/4)d = a

(3/4)c + (3/4)d = b => (1/4)c + (1/4)d = b/3 => a = b/3

b = c

a = d => d = b/3

a + b + c + d = 1

=> (b/3) + b + b + (b/3) = 1

=> 8b/3 = 1

=> b = 3/8

Thus, c = 3/8

d = a = b/3 = 1/8

Thus,

W = [1/8 , 3/8 , 3/8 , 1/8]

Second question,

Let the states H, A and L denote high, average and low ratings.

The transition probability matrix is,

Probability that ratings is average the second week given high this week = P(X2 = A | X0 = H)  

= P(X2 = A , X1 = H, X0 = H) +  P(X2 = A , X1 = A, X0 = H)  

= 0.8 * 0.2 + 0.2 * 0.1 = 0.18

Probability that ratings is average the second week given high the first week = P(X2 = A | X1 = H) = 0.1

Probability that ratings is low the third week given average first week = P(X3 = L | X1 = A)  

= P(X3 = L , X2 = H, X1 = A) +  P(X3 = L , X2 = A, X1 = A)

= 0.1 * 0 + 0.1 * 0.8 = 0.08