1. (45 total points)

For a stationary Discrete-Time Markov Chain (DTMC) we have, in standard form, P, P2,
and P3, P∞ and the associated matrix (I − Q)−1. The state space for this system is the set
{1, 2, 3, 4, 5}

(a) (5 points): If the initial state at time 0 is 3, and three thousand transitions later the
state is 1, what is the probability that the process enters a transient state at epoch 3002?

(b) (5 points): If the process starts in State 1, what is the expected number of visits to State
3 in the first 3 epochs?

(c) (5 points): Assuming that at epoch 3, the system is in State 3, compute the probability
that the system ever visits State 4 in two ways?

(d) (5 points): What is the mean number of epochs spent in State 2, per visit to State 2?

(e) (5 points): Will solving the set of simultaneous linear equations Π = Π × P, along with
the normalizing equation Π × 1 = 1 provide a meaningful solution? If so, specify the
algorithm and offer an interpretation for the resulting vector Π, and if not, then explain
why not. Here 1 is a 5 × 1 column vector.

(f) (5 points): If the process starts at time zero in State 1, what is the mean time until the
process enters its terminating state?

(g) (5 points): Since this P is already written in standard form, specify the R submatrix
(actual values are necessary here), and interpret the meaning of R.

(h) (10 points): In the long-run, what is the variance of the time until the first visit to State
4 from initial state State 1, if we know that State 4 will definitely be visited? Write out
the steps of the your algorithm and then compute the exact number using MATLAB
and any MATLAB code that I provided in the class.

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