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3. Let {X_{n} : n ? 0} be a Markov chain with state space {0, 1, 2, 3} and transition matrix P = egin{pmatrix} 1/4 & 0 & 1/2 & 1/4 \ 0 & 1/5 & 0 & 4/5 \ 0 & 1 & 0 & 0 \ 1/3 & 1/3 & 0 & 1/3 end{pmatrix}. a. Find the irreducible classes of intercommunicating states, and classify them in terms of positive recurrence, null recurrence, transience, and periodicity. b. Find P(X_{7} = 2, X_{5} = 0, X_{3} = 0 | X_{0} = 2). c. Given that underline{p}^{(0)} = (0.25, 0.5, 0, 0.25), find the distribution of the location of the chain at time 2. d. Find the probability that the chain visits state 3 before visiting state 1, given X_{0} = 0.

          3. Let {X_{n} : n ? 0} be a Markov chain with state space {0, 1, 2, 3} and transition matrix

P = egin{pmatrix} 1/4 & 0 & 1/2 & 1/4 \ 0 & 1/5 & 0 & 4/5 \ 0 & 1 & 0 & 0 \ 1/3 & 1/3 & 0 & 1/3 end{pmatrix}.

a. Find the irreducible classes of intercommunicating states, and classify them in terms of positive recurrence, null recurrence, transience, and periodicity.
b. Find P(X_{7} = 2, X_{5} = 0, X_{3} = 0 | X_{0} = 2).
c. Given that underline{p}^{(0)} = (0.25, 0.5, 0, 0.25), find the distribution of the location of the chain at time 2.
d. Find the probability that the chain visits state 3 before visiting state 1, given X_{0} = 0.

3. Let Xn : n ? 0 be a Markov chain with state space 0, 1, 2, 3 and transition matrix

P = eginpmatrix 1/4 0 1/2 1/4 0 1/5 0 4/5 0 1 0 0 1/3 1/3 0 1/3 endpmatrix.

a. Find the irreducible classes of intercommunicating states, and classify them in terms of positive recurrence, null recurrence, transience, and periodicity.
b. Find P(X7 = 2, X5 = 0, X3 = 0 | X0 = 2).
c. Given that underlinep^(0) = (0.25, 0.5, 0, 0.25), find the distribution of the location of the chain at time 2.
d. Find the probability that the chain visits state 3 before visiting state 1, given X0 = 0.

Added by Jennifer H.

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