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We consider a three-state Markov chain with transition matrix \(\mathbf{Q}\) (states A, B, and C):$$ \mathbf{Q} = \begin{pmatrix} 0.2 & 0 & 0.8 \\ 0.1 & 0.3 & 0.6 \\ 0.5 & 0.5 & 0 \end{pmatrix} $$The initial probability distribution is \(\pi_0 = \begin{pmatrix} 0.5 & 0.5 & 0 \end{pmatrix}\).
Give the values of \(P(X_0 = A)\), \(P(X_0 = B)\), and \(P(X_0 = C)\).
Calculate the probability vectors \(\pi_1\) and \(\pi_2\).
Deduce the values of \(P(X_1 = A)\) and \(P(X_2 = C)\).
Express \(\pi_n\) in terms of \(\pi_0\) and \(\mathbf{Q}\).
Using a calculator, find \(\pi_{10}\) and \(\pi_{20}\). How does the distribution evolve?
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