Question

Consider a finite population (of fixed size \(N\) ) of individuals of possible types \(A\) and \(a\) undergoing the following growth process. At instants of time \(t_{1}

Short answer

The transition probabilities for the Markov chain {X_n} are: - P(j to j+1) = \( (\frac{(N-j) \mu_2}{B_j}) (\frac{j}{N}) \) - P(j to j-1) = \( (\frac{j \mu_1}{B_j}) (\frac{N-j}{N}) \)

Step-by-step solution

Analyze given probabilities

At time \(t_n\), we have j A's and (N-j) a's in the population. - Probability that an A individual dies := \( \frac{j \mu_1}{B_j}\) - Probability that an a individual dies := \( \frac{(N-j) \mu_2}{B_j}\) Where \(B_j = \mu_1 j + \mu_2 (N-j)\). - Probability that the new individual is A := \( \frac{j}{N}\) - Probability that the new individual is a := \( \frac{N-j}{N}\)

Compute transition probabilities

Recall that the transition probability is the probability of moving from one state to another in a Markov chain. In this case, we want to find the probability of moving from j A's to j+1 A's, and from j A's to j-1 A's in one step.

Step 2a: Transition probability from j A's to j+1 A's

To get to j+1 A's, an a individual must die, and the new individual must be an A. - Probability of an a dying = \( \frac{(N-j) \mu_2}{B_j}\) - Probability of new individual being A = \( \frac{j}{N}\) So, the transition probability P(j to j+1) := \( (\frac{(N-j) \mu_2}{B_j}) (\frac{j}{N}) \)

Step 2b: Transition probability from j A's to j-1 A's

To get to j-1 A's, an A individual must die, and the new individual must be an a. - Probability of an A dying = \( \frac{j \mu_1}{B_j}\) - Probability of new individual being a = \( \frac{N-j}{N}\) So, the transition probability P(j to j-1) := \( (\frac{j \mu_1}{B_j}) (\frac{N-j}{N}) \)

Write down the transition probabilities

The transition probabilities for the Markov chain {X_n} are: - P(j to j+1) = \( (\frac{(N-j) \mu_2}{B_j}) (\frac{j}{N}) \) - P(j to j-1) = \( (\frac{j \mu_1}{B_j}) (\frac{N-j}{N}) \)