midterm_2015_solutions - MS&E 221 Ramesh Johari Midterm...

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MS&E 221 Midterm Examination Ramesh Johari Feb. 9, 2015 Instructions 1. The exam is due by 2:00 PM on February 11, 2015 . No exceptions to this deadline will be made, except for medical necessity. 2. If you hand in your exam on paper, turn it into the homework box in the basement of Huang Engineering Center. 3. You may use any textbooks and the lecture notes from the class as resources, as well as MAT- LAB (or equivalent software); however, you may not collaborate with any other students on the exam, and you may not use any online resources. 4. Include a signed acknowledgment of the honor code (below). 5. If you e-mail your exam, please send it to Apaar: [email protected] . Include an acknowledgment of the honor code (below) in your e-mail. Put “221 midterm” in your subject line. 6. The exam will be scored out of 70 points. 7. You will receive partial credit, so please show your work. However, note that any incorrect answer will be marked down accordingly. Honor Code In taking this examination, I acknowledge and accept the Stanford University Honor Code. NAME (signed) NAME (printed) 1
Problem 1 (20 points; 5 points per part). Suppose Z 0 , Z 1 , Z 2 , . . . are i.i.d. random variables that represent the demand of a certain product in time period n ; assume each Z n takes values in the nonnegative integers. In each part below, we give a definition for X n , n 0 , as a particular summary statistic of Z 0 , . . . , Z n . In each case, is X 0 , X 1 , X 2 , . . . always a Markov chain? Explain your answer. If X n is not necessarily a Markov chain, suggest how the state space could be expanded to convert it into a Markov chain. (a) (Cumulative demand) X n = Z n + Z n - 1 + . . . + Z 0 .
(b) (Maximum demand) X n = max { Z n , Z n - 1 , . . . , Z 0 } .
(c) (Average recent demand) For n k , X n = Z n + Z n - 1 + . . . + Z n - k k + 1 , where k is a fixed positive integer. (For n < k we define X n = 0 .)