Question

Consider the Markov chain below. Let us refer to a transition that results in a state with a higher (respectively, lower) index as a birth (respectively, death). Calculate the following probabilities, assuming that when we start observing the chain, it is already in steady-state.

1. The steady-state probabilities for each state.
pi_1 = 0.2
pi_2 = 0.4
pi_3 = 0.4

2. The probability that the first transition we observe is a birth.
Unanswered

3. The probability that the first change of state we observe is a birth.
Unanswered

Answer 1

2. 0.2
3. 9/25

Answer 2

To calculate the probability that the first transition we observe is a birth, we need to consider the transition probabilities from each state.

Let's denote the transition probabilities as P(i,j), where i is the current state and j is the next state.

In this case, the only possible birth transition is from state 1 to state 2, as state 1 is the only state with a higher index.

Thus, the probability of observing a birth as the first transition can be calculated as:

P(1,2) = pi_1 * P(1,2) = 0.2 * P(1,2)

Similarly, to calculate the probability that the first change of state we observe is a birth, we need to consider all possible birth transitions from each state.

The possible birth transitions in this case are:
- From state 1 to state 2
- From state 2 to state 3

Thus, the probability of observing a birth as the first change of state can be calculated as:

P(1,2) + P(2,3) = pi_1 * P(1,2) + pi_2 * P(2,3) = 0.2 * P(1,2) + 0.4 * P(2,3)

To calculate these probabilities, we need to know the specific transition probabilities (P(i,j)) in the Markov chain.

Answer 3

Thank you for providing the specific transition probabilities.

2. The probability that the first transition we observe is a birth can be calculated as:
P(1,2) = pi_1 * P(1,2) = 0.2 * 0.2 = 0.04

So, the probability that the first transition we observe is a birth is 0.04 or 4%.

3. The probability that the first change of state we observe is a birth can be calculated as:
P(1,2) + P(2,3) = pi_1 * P(1,2) + pi_2 * P(2,3) = 0.2 * 0.2 + 0.4 * 0.4 = 0.04 + 0.16 = 0.2

So, the probability that the first change of state we observe is a birth is 0.2 or 20%.