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finalS09 - IEOR 4106 Introduction to Operations Research...

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IEOR 4106: Introduction to Operations Research: Stochastic Models Spring 2009, Professor Whitt, Final Exam Chapters 4-7 and 10 in Ross, Tuesday, May 12, 9:00am-12:00noon Open Book: but only the textbook, the CTMC notes and three pages of notes Honor Code: Students are expected to behave honorably when taking exams. After completing the test, please testify to your adherence by writing the following on your bluebook and signing your name: “I have neither given nor received help while taking this test.” Justify your answers; show your work. 1. Oatpower Stock (27 points) Ever since the 2003 power outage in the northeastern United States, there has been grow- ing investor enthusiasm for the company Oatpower, Inc., which is developing a new way to efficiently generate vast power from ordinary oats. Oatpower claims that it will be possible to generate sufficient power from a single cup of oats to run a subway train for ten years. If Oatpower is successful, subways and elevators will no longer have to depend on America’s aging electric power grid. The power generation method is highly secret, but there is a rumor that it is based on a surprising chemical reaction between oats and Raspberry Snapple. The current price of Oatpower stock is $70 per share. Suppose that the Oatpower stock price over time (measured in years) can be modelled as the stochastic process X ≡ { X ( t ) : t 0 } , where X ( t ) 70 + 10 B ( t ) , t 0 , with B ≡ { B ( t ) : t 0 } being standard (drift zero, unit variance) Brownian motion. Let M ≡ { M ( t ) : t 0 } be the associated stochastic process defined by M ( t ) max 0 s t { X ( s ) } , t 0 . For any random variable Z , let V ar ( Z ) be the variance of Z . Calculate the following quantities: (a) P ( X (4) > 90); (b) V ar ( X (1) + X (4)); (c) E [ X (4) | X (1) = 80]; (d) E [ X (2) X (4) | X (1) = 80]; (e) E [ X (1) | X (4) = 90]; (f) P ( M (4) > 90); (g) P ( M (4) 70).
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