https://file.aiquickdraw.com/mathbot/file/1726680979125p6eb5tuv.pdf

Let's solve the problems presented in the document:

### 1. **Classify the States of a Markov Chain**

Given the **transition probability matrix (TPM):**
\[
\begin{pmatrix}
0 & 1 \\
\frac{1}{2} & \frac{1}{2}
\end{pmatrix}
\]

This represents a Markov chain with two states: 0 and 1.

- **State 0**:
  - From state 0, you always move to state 1. Thus, state 0 is transient (you don't return to it once you leave).
  
- **State 1**:
  - From state 1, there's a 50% chance of returning to state 1 and a 50% chance of moving to state 0. This implies that you can return to state 1, making it recurrent.

Thus, **state 0 is transient**, and **state 1 is recurrent**.

---

### 2. **Find the Steady State Distribution**

We need to solve for the steady state distribution \(\pi = (\pi_0, \pi_1)\), where the steady state distribution satisfies:
\[
\pi P = \pi
\]
for the given transition matrix:
\[
P = 
\begin{pmatrix}
0 & 1 \\
\frac{1}{2} & \frac{1}{2}
\end{pmatrix}
\]

This gives the following system of equations:
\[
\pi_0 = \frac{1}{2} \pi_1
\]
\[
\pi_1 = \pi_0 + \frac{1}{2} \pi_1
\]

Also, we know that \(\pi_0 + \pi_1 = 1\). Using this, we solve the equations to find the steady-state distribution.

---

### 3. **Probability Distribution after 3 Steps**

Given the initial state probability distribution \(P(0) = \left( \frac{2}{3}, \frac{1}{3} \right)\) and the same transition matrix \(P\), we calculate the distribution after 3 steps using:
\[
P(3) = P(0) P^3
\]
We can compute this using matrix multiplication.

---

### 4. **Steady State Distribution of Another TPM**

Given the new transition matrix:
\[
P = 
\begin{pmatrix}
0.4 & 0.6 & 0 & 0 \\
0.3 & 0.7 & 0 & 0 \\
0.2 & 0.4 & 0.1 & 0.3 \\
0 & 0 & 0 & 1
\end{pmatrix}
\]

We need to find the steady-state distribution by solving the system \(\pi P = \pi\), along with the normalization condition \(\pi_1 + \pi_2 + \pi_3 + \pi_4 = 1\).

Let me know which problem you'd like to go through in more detail!

This page provides a step-by-step solution to classifying states and finding the steady-state distribution of Markov chains. Learn how to compute the steady state and probability distribution after multiple steps using transition probability matrices (TPM). Suitable for B.Tech. students in computational statistics.

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