midterm_2008

# midterm_2008 - MS&E 221 Ramesh Johari Midterm Examination...

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MS&E 221 Midterm Examination Ramesh Johari February 13, 2006 Instructions 1. Take alternate seating if possible. 2. Answer all questions in the spaces provided on these sheets. If needed, additional paper will be available at the front of the room. Answers given on any other paper will not be counted. 3. The examination begins at 11:05 am, and ends at 12:15 pm. 4. No notes, books, or calculators are allowed. 5. The exam will be scored out of 100 points. 6. Pace yourself! Honor Code In taking this examination, I acknowledge and accept the Stanford University Honor Code. NAME (signed) NAME (printed) 1

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Problem 1 (30 points). (Short answer) Answer each of the following questions. (a) (10 points) Suppose { X n } is an irreducible Markov chain with transition matrix P and invariant distribution π . Find the long run fraction of time that transitions from state i to state j are observed. (b) (10 points) Suppose in a given Markov chain, states i and j communicate with each other. Is it always true that lim n →∞ P ( X n = i | X 0 = j ) > 0 ? If true, justify your answer; if not, provide a counterexample. (c) (10 points) Let { X n } and { Y n } be two irreducible Markov chains oxn { 0 , 1 , 2 , . . . , N } that are independent of each other. Let T
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midterm_2008 - MS&E 221 Ramesh Johari Midterm Examination...

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