A Markov chain (left{X_{0}, X_{1}, ldots ight}) has state space (mathbf{Z}={0,1,2}) and transition matrix [mathbf{P}=left(begin{array}{ccc} 0.5 &

A Markov chain \(\left\{X_{0}, X_{1}, \ldots\right\}\) has state space \(\mathbf{Z}=\{0,1,2\}\) and transition matrix

\[\mathbf{P}=\left(\begin{array}{ccc} 0.5 & 0 & 0.5 \\ 0.4 & 0.2 & 0.4 \\ 0 & 0.4 & 0.6 \end{array}\right)\]

(1) Determine \(P\left(X_{2}=2 \mid X_{1}=0, X_{0}=1\right)\) and \(P\left(X_{2}=2, X_{1}=0 \mid X_{0}=1\right)\).

(2) Determine \(P\left(X_{2}=2, X_{1}=0 \mid X_{0}=0\right)\) and, for \(n>1\),

\[P\left(X_{n+1}=2, X_{n}=0 \mid X_{n-1}=0\right)\]

(3) Assuming the initial distribution

\[P\left(X_{0}=0\right)=0.4 ; P\left(X_{0}=1\right)=P\left(X_{0}=2\right)=0.3\]

determine \(P\left(X_{1}=2\right)\) and \(P\left(X_{1}=1, X_{2}=2\right)\).