Quiz question For each of the following transition matrices for Markov chains with state spaceS = {1, 2}, write down the Full Balance Equations and find the equilibrium distribution(s).(a) P =0.4 0.60.8 0.2(b) P =0.82 0.180.82 0.18

Sure, let's solve this step by step.

(a) For the transition matrix P = [0.4 0.6; 0.8 0.2], the Full Balance Equations are:

For state 1: π1 = 0.4π1 + 0.8π2 For state 2: π2 = 0.6π1 + 0.2π2

We also know that π1 + π2 = 1 (since the probabilities must add up to 1)

Solving these equations, we get π1 = 4/7 and π2 = 3/7. So, the equilibrium distribution is (4/7, 3/7).

(b) For the transition matrix P = [0.82 0.18; 0.82 0.18], the Full Balance Equations are:

For state 1: π1 = 0.82π1 + 0.82π2 For state 2: π2 = 0.18π1 + 0.18π2

We also know that π1 + π2 = 1 (since the probabilities must add up to 1)

Solving these equations, we get π1 = 1/2 and π2 = 1/2. So, the equilibrium distribution is (1/2, 1/2).

Sure, let's solve this step by step.

(a) For the transition matrix P = [0.4 0.6; 0.8 0.2], the full balance equations are:

π1 * P11 + π2 * P21 = π1 π1 * P12 + π2 * P22 = π2

Substituting the values from the matrix, we get:

π1 * 0.4 + π2 * 0.8 = π1 π1 * 0.6 + π2 * 0.2 = π2

We also know that π1 + π2 = 1 (since the probabilities must add up to 1).

Solving these equations, we get π1 = 2/3 and π2 = 1/3. So, the equilibrium distribution is (2/3, 1/3).

(b) For the transition matrix P = [0.82 0.18; 0.82 0.18], the full balance equations are:

π1 * P11 + π2 * P21 = π1 π1 * P12 + π2 * P22 = π2

Substituting the values from the matrix, we get:

π1 * 0.82 + π2 * 0.82 = π1 π1 * 0.18 + π2 * 0.18 = π2

We also know that π1 + π2 = 1 (since the probabilities must add up to 1).

Solving these equations, we get π1 = 1/2 and π2 = 1/2. So, the equilibrium distribution is (1/2, 1/2).