Question

A two state Markov chain has the transition probability matrix p = [0 0 1 i1 a b a 1-b]. (a) Determine the first return distribution. (b) Verify equation (3.2) when i = 0. (Refer to iii, (5.2)).

Answer 1

To determine the first return distribution for a two-state Markov chain, find the stationary distribution by solving the equation π = πP. The first return distribution is then the probability of returning to State 1 first, given that the chain starts in State 1.

Step-by-step explanation:

To determine the first return distribution for a two-state Markov chain with the given transition probability matrix, we need to find the stationary distribution. A stationary distribution is a probability distribution that remains unchanged after each transition. To find it, we solve the equation π = πP, where π is the stationary distribution and P is the transition probability matrix.

Substituting the given matrix into the equation, we have:

π = [π 0] [0 0 1 i1 a b a 1-b]

Simplifying the equation, we get:

π = [πa π(1-b)]

We can solve this system of equations to find the values of π.

Once we have the stationary distribution, the first return distribution is simply the probability that the chain returns to State 1 first, given that it starts in State 1. This is the first entry in the stationary distribution.

In equation (3.2), when i = 0, it refers to the probability of transitioning from State 0 to State 0 in one step. We can calculate this probability using the entry in the transition probability matrix corresponding to this transition.