Question

In example 4.13, give the transition probabilities of the yn Markov chain in terms of the transition probabilities pi,j of the xn chain.

A) P(yn) = P(xn)
B) P(yn) = P(xn) / P(xn-1)
C) P(yn) = ∑ P(xn, xn-1)
D) P(yn) = P(xn) * P(xn-1)

Answer 1

The question is about finding the transition probabilities of the yn Markov chain in terms of the xn chain. Without additional context or information, it's not possible to provide a definitive answer. However, calculating conditional probabilities in general involves the formula P(A AND B) / P(B), and the independence of events alters the addition rule for probabilities.

Step-by-step explanation:

The student's question pertains to the transitioning probabilities of a Markov chain, and specifically how to express the transition probabilities of the yn Markov chain in terms of the transition probabilities pi,j of the xn chain. To answer this, we would look for a mathematical relationship between the two chains' transition probabilities. However, the provided information does not give a clear way to relate yn to xn, thus we cannot confidently provide one of the options (A, B, C, D) as the correct answer.

To calculate a conditional probability like P(A|B), where A and B are events, you use the formula P(A AND B) / P(B), assuming that P(B) is greater than zero. If events Y and Z are independent, the probability of Y OR Z occurring is P(Y) + P(Z) - P(Y)P(Z).

Answer 2

The transition probabilities Q_{i,j} of the Y_n Markov chain in Example 4.13 are not explicitly provided, and additional information is needed to determine them accurately.

Step-by-step explanation:

In the context of Example 4.13, the transition probabilities of the \(Y_n\) Markov chain (\(Q_{i,j}\)) are not explicitly provided. However, based on the description, we can infer the following:

Q_i,j = P(Y_{n+1} = j ∣ Y_n = i)

1. Q_{1,4} : Probability of transitioning from state 1 to state 4, indicating the pattern has appeared.

2. Q_{2,5} : Probability of transitioning from state 2 to state 5, indicating progress toward the pattern when the current state is 2.

3. Q_{3,6}: Probability of transitioning from state 3 to state 6, indicating no progress when the current state is 3.

4. Q_{4,4} : Probability of staying in state 4, as it is an absorbing state.

Therefore, the transition probabilities in terms of the X_n chain are not directly provided in the given context, and more information would be needed to compute them precisely.