Part b
Given the initial population distribution \(v_0 = \begin{pmatrix} 0.3 \\ 0.4 \\ 0.3 \end{pmatrix}\), use the transition matrix \(A\) from part (a) to compute the population distribution after 5 years, that is, calculate \(A^5 \times v_0\). Round your answer to three decimal places.
Population Migration via Markov Chains
A country is divided into three regions: North, Central, and South. Each year, a percentage of the population migrates from one region to another according to the following data: – From North: 80% remain in North, 10% move to Central, and 10% move to South. – From Central: 20% move to North, 50% remain in […]
Absorbing Markov Chain Analysis
Consider a Markov chain model representing a system with two transient states (State 1 and State 2) and one absorbing state (State 3). The transition probabilities between the transient states are given by the submatrix \( Q \), and the transition probabilities from the transient states to the absorbing state are given by the matrix […]
Part b
Using the iterative method, compute an approximate stationary distribution. Start from \(\pi^{(0)} = (0.5, 0.5)\) and perform at least 5 iterations using the relation \(\pi^{(n+1)} = \pi^{(n)} P\).
Stationary Vector Verification
Given the transition matrix \(P = \begin{pmatrix} 0.6 & 0.4 \\ 0.3 & 0.7 \end{pmatrix}\) and the candidate stationary distribution vector \(v = \begin{pmatrix} 0.5 \\ 0.5 \end{pmatrix}\), verify whether \(v\) is a stationary vector for \(P\). Show that multiplying \(P\) by \(v\) yields \(v\).
3-Step Transitions in a 3-State Markov Chain
Consider a 3-state Markov chain with transition matrix \(P\). To find the 3-step transition probabilities, explain the process to compute \(P^3\) and state one key property of Markov chains that ensures the validity of the resulting probability matrix.
Calculating 2-Step Transition Probabilities Using Matrix Multiplication
Given a transition matrix \(P\) for a Markov Chain: \[ P = \begin{bmatrix}0.7 & 0.3 \\ 0.4 & 0.6 \end{bmatrix} \] Calculate the probability of transitioning from state 1 to state 2 in exactly 2 steps. Show all steps of your calculation.
2-Step Transition Matrix Probabilities
Explain how matrix powers are used to calculate 2-step transition probabilities for a Markov chain. In your answer, describe what the entry \((T^2)_{ij}\) represents.
Analyzing Markov Chain Transition Behavior
Consider a Markov chain with transition matrix \(M = \begin{pmatrix}0.8 & 0.2\\0.3 & 0.7\end{pmatrix}\). Given an initial state distribution \(\vec{x_0} = (1, 0)\), (a) Calculate the state distribution after 3 transitions using repeated multiplication. (b) Discuss the long-term trend of the state distributions as the number of transitions increases.
Part a
Compute the two-step transition matrix \( P^2 \) for the Markov chain with \( P = \begin{pmatrix} 0.8 & 0.2 \\ 0.3 & 0.7 \end{pmatrix} \). Show all steps in your calculation.