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Unformatted text preview: Math 4740 Exam 2, March 30, 2009 R. Durrett 1. (21 points) Let Y 1 ,Y 2 ,... be independent with P ( Y i = 2) = p and P ( Y i = 1 / 2) = 1 p . Let X n = X Y 1 Â·Â·Â· Y n and think of X n as the price of a stock at time n . (a) What choice of p makes X n a martingale? We will use this value of p for the rest of the problem, and suppose X = 1. (b) Find lim n â†’âˆž (1 /n ) log 2 X n (log base 2) and conclude X n â†’ 0. (c) Find P ( T 4 < âˆž ) where T 4 = min { n â‰¥ 0 : X n = 4 } . 2. (20 points) When Chad arrives at the bank, Al and Bob are being served by tellers 1 and 2, who have exponential service times with means 4 and 2 minutes. When the first service is complete Chad goes to the teller that is free. (a) Find the probabilities that Al, Bob and Chad are the last to leave. (b) What is the expected time Chad is at the bank (waiting plus service time)? 3. (39 points) A policewoman on the evening shift writes a Poisson mean 6 number of tickets per hour. 2/3â€™s of these are for speeding and cost $100....
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 Spring '10
 DURRETT
 Math, Standard Deviation, Mean, Probability theory, Xn, time Chad

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