Convergence Analysis Using Markov Chains

00:00 7 marks

Consider a Markov chain with the transition matrix \(P = \begin{pmatrix}0.7 & 0.3 \\ 0.4 & 0.6\end{pmatrix}\) and an initial distribution \(\pi^{(0)} = (0.5, 0.5)\). Perform the following:

(a) Calculate the distribution after 3 steps.

(b) Evaluate the difference between the distribution after 3 steps and the steady-state distribution obtained by solving \(\pi P = \pi\).

Part a

7 minutes 4 marks

Question

Given a transition matrix \(P = \begin{pmatrix}0.7 & 0.3 \\ 0.4 & 0.6\end{pmatrix}\) and an initial distribution \(\pi^{(0)} = (0.5, 0.5)\), calculate the distribution after 3 steps. Show your working.

Hint Hint

Hint

Multiply the transition matrix by the distribution vector repeatedly to compute \(\pi^{(3)}\).

Answer Answer
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Answer Answer

Complete Answer

The distribution after 3 steps is approximately \(\pi^{(3)} \approx (0.570, 0.430)\).

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Part b

5 minutes 3 marks

Question

Explain the condition for a Markov chain to reach a steady state. Calculate the difference between the distribution after 3 steps and the steady-state distribution obtained by solving \(\pi P = \pi\).

Hint Hint

Hint

Solve \(\pi P = \pi\) to find the steady state and compare it with \(\pi^{(3)}\).

Answer Answer
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Explanation Explanation
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Answer Answer

Complete Answer

The steady-state distribution is approximately \(\pi \approx (0.571, 0.429)\), and the difference \(\|\pi^{(3)} – \pi\|\) is about \(0.002\), indicating near convergence.

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