https://file.aiquickdraw.com/mathbot/file/17285882186190yyp3uz7.jpg

Let's break down the Markov chain problem from the image and solve it step by step.

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### Given: 
Markov chain \( X_n \) with states \( \{1, 2, 3\} \) and the transition probability matrix:

\[
P = \begin{pmatrix}
0.4 & 0.5 & 0.1 \\
0.2 & 0.3 & 0.5 \\
0.3 & 0.4 & 0.3
\end{pmatrix}
\]

This matrix describes the probability of transitioning from one state to another. For example, the probability of moving from state 1 to state 2 is 0.5, and from state 2 to state 3 is 0.5.

### (a) Finding \( E[X_2] \) given the initial distribution \( p_1 = p_2 = \frac{1}{4}, p_3 = \frac{1}{2} \):

The expected value of \( X_2 \) depends on the initial distribution and the transition matrix after two steps.

- Start by finding the distribution after two steps, \( P(X_2 = i) \), by calculating the matrix \( P^2 \) (i.e., the matrix for two transitions). Then, apply this to the initial distribution vector \( \mathbf{p} = \begin{pmatrix} \frac{1}{4} \\ \frac{1}{4} \\ \frac{1}{2} \end{pmatrix} \).

### (b) Finding \( P(X_n = 3 \text{ for some } n \leq 4 \mid X_0 = 1) \):

This is asking for the probability that, starting in state 1, the chain enters state 3 at least once within the first four steps. This involves finding the cumulative probability of entering state 3 in 1, 2, 3, or 4 steps. We can calculate this by evaluating the probabilities \( P(X_1 = 3), P(X_2 = 3), P(X_3 = 3), P(X_4 = 3) \).

### (c) Given a large number of transitions, finding the probability that the previous state was 1, given that the current state is 2:

After a large number of transitions, the system reaches its **stationary distribution**. We first need to compute the stationary distribution vector \( \pi = (\pi_1, \pi_2, \pi_3) \), which satisfies \( \pi P = \pi \). Then, use Bayes' theorem to compute \( P(X_{n-1} = 1 \mid X_n = 2) \) based on the stationary distribution and the transition matrix.

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Would you like a detailed solution to any of the specific parts (a, b, or c) now? Let me know if you have any questions! Here are 5 relevant questions:

1. What is the general approach to compute powers of a transition matrix?
2. How do we find the expected value in Markov chains based on the transition matrix?
3. What role does the stationary distribution play in long-term predictions in Markov chains?
4. How can we compute the probability of reaching a particular state within a given number of steps?
5. What is the significance of the initial state distribution in Markov chain problems?

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**Tip**: When working with Markov chains, understanding the power of the transition matrix helps in analyzing multiple-step transitions.

Consider a Markov chain X_n, n ≥ 0 with states 1, 2, 3 and transition probability matrix as shown, and solve the following: (a) If P(X_0 = i) = p_i, where p_1 = p_2 = 1/4, p_3 = 1/2, find E[X_2]. (b) Find P(X_n = 3 for some n ≤ 4 | X_0 = 1). (c) Given a large number of transitions, if the current state is 2, what is the probability that the previous state was 1?

Solve a Markov chain problem with states 1, 2, 3 and a given transition matrix. Learn to find E[X_2] given initial probabilities, calculate the probability of reaching state 3 within four transitions, and use the stationary distribution to find the probability of previous state in a long sequence.

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