Math Problem Statement
HW (1).pdf
289.37 KB
solve it
Solution
Let's solve the problems presented in the document:
1. Classify the States of a Markov Chain
Given the transition probability matrix (TPM):
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!Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Markov Chains
Probability Theory
Steady State Distribution
Formulas
Steady state distribution equation: πP = π
Transition probability matrix (TPM)
Theorems
Classification of states in Markov chains
Steady state theorem
Suitable Grade Level
Undergraduate (B.Tech.)
Related Recommendation
Find the Steady-State Distribution Vector for a Transition Matrix
Markov Chain with Transition Probability Matrix and Initial Probabilities
Find the Steady-State Vector of Matrix P = [[0.4, 0.5], [0.6, 0.5]]
Markov Chain with Transition Matrix and Steady-State Distribution
Transition Probability Matrix for Demographic Regions