Problem 5.3. A trained mouse lives in the house shown in the Fig. 5.4. A bell rings at regular intervals (short compared to the mouses lifetime). Each time it rings, the mouse changes rooms. When he changes rooms, he is equally likely to pass through any of the doors of the room he is in. Let the stochastic variable Y denote "mouse in a particular room." There are three realizations of Y: "mouse in room A," "mouse in room B," and "mouse in room C," which we denote as y(1), y(2), and y(3), respectively.

(a) Compute the transition matrix, Q, and the conditional probability matrix, P(sols).

(b) Compute the probability vector, (P(s), at time s, given that the mouse starts in room C. Approximately what fraction of his life does he spend in each room?

(c) Assume that the realization, y(n), equals n. Compute the first moment, (y(s)), and the autocorrelation function, (y(0)y(s)), for the same initial conditions as in part (b).