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midterm_2006

# midterm_2006 - 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 1:05 pm, and ends at 2:15 pm. 4. No notes, books, or calculators are allowed. 5. The exam will be scored out of 150 points. 6. Pace yourself! Note that part 2(f) and part 3(e)-(f) are harder than the rest of the exam, and you should leave them for last. 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 (40 points). Answer each of the following questions TRUE or FALSE. If TRUE, provide a brief justification (1-2 sentences) for your answer; if FALSE, provide a brief counterexample. (5 points per correct answer; 5 points per correct justification/counterexample) (a) Every Markov chain with a finite state space has at least one closed class. (b) Every Markov chain on a finite state space with a symmetric transition matrix has an invariant distribution. (A matrix is symmetric if A ij = A ji for all i, j .) 2
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