MATH 632 Midterm: 2003 Math 632 - Fall Exam 1
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Instructions: show calculations and justify non-obvious statements for full credit. When you quote a result proved in class, state the hypotheses and conclusion clearly, and justify why the hypotheses are met. 1/3 2/3 0 (a) (15 pts) explain why the power pn of the matrix converges as n and nd the limit matrix. (b) (10 pts) let xn be a markov chain with transition matrix p given above. Let be the invariant distribution for this chain. P (cid:2)x7 = 1, x12 = 1(cid:3). (c) (10 pts) find p [xt2+2 = 2], the probability that 2 steps after the. 1 (a) (10 pts) draw the arrow diagram for the markov chain. Classify the states according to recurrence and transience. Find the periods of recurrent states. (be careful here because your subsequent answers depend on this part. ) (b) (15 pts) find the probabilities 3,4 and 1,4, and the expectations.
