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MIT 6 041 - Probabilistic Systems Analysis

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6.041/6.431 Fall 2010 Final Exam Wednesday, December 15, 9:00AM - 12:00noon. DO NOT TURN THIS PAGE OVER UNTIL YOU ARE TOLD TO DO SO Name: Recitation Instructor: TA: Question Score Out of 1.1 4 1.2 4 1.3 4 1.4 4 1.5 4 1.6 4 1.7 4 1.8 4 2.1 3 2.2 (a) 3 2.2 (b) 3 2.2 (c) 3 Question Score Out of 2.3 (a) 3 2.3 (b) 3 2.4 (a) 5 2.4 (b) 5 2.4 (c) 5 2.4 (d) 5 2.5 5 2.6 5 2.7 5 2.8 5 2.9 (a) 5 2.9 (b) 5 Your Grade 100 • Show your work and provide brief justifications for your answers, except for parts where you are told that a justification is not needed. • Unless instructed otherwise, for full credit answers should be algebraic ex-pressions (no integrals), in simplified form. These expressions may involve constants such as π or e, and need not be evaluated numerically. • This quiz has 2 problems, worth a total of 100 points. • You may tear apart page 3 and 4, as per your convenience, but you must turn them in together with the rest of the booklet. • Write your solutions in this quiz booklet, only solutions in this quiz booklet will be graded. Be neat! You will not get credit if we can’t read it. • You are allowed three two-sided, handwritten, 8.5 by 11 formula sheets. Calculators are not allowed.Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) Problem 1. (32 points) Consider a Markov chain {Xn; n = 0, 1, . . .}, specified by the following transition diagram. 0.60.90.30.10.20.40.51 2 31. (4 points) Given that the chain starts with X0 = 1, find the probability that X2 = 2. 2. (4 points) Find the steady-state probabilities π1, π2, π3 of the different states. In case you did not do part (b) correctly, in all subsequent parts of this problem you can just use the symbols πi: you do not need to plug in actual numbers. 3. (4 points) Let Yn = Xn − Xn−1. Thus, Yn = 1 indicates that the nth transition was to the right, Yn = 0 indicates it was a self-transition, and Yn = −1 indicates it was a transition to the left. Find lim P(Yn = 1). n→∞ 4. (4 points) Is the sequence Yn a Markov chain? Justify your answer. 5. (4 points) Given that the nth transition was a transition to the right (Yn = 1), find the probability that the previous state was state 1. (You can assume that n is large.) 6. (4 points) Suppose that X0 = 1. Let T be defined as the first positive time at which the state is again equal to 1. Show how to find E[T ]. (It is enough to write down whatever equation(s) needs to be solved; you do not have to actually solve it/them or to produce a numerical answer.) 7. (4 points) Does the sequence X1, X2, X3, . . . converge in probability? If yes, to what? If not, just say “no” without explanation. 8. (4 points) Let Zn = max{X1, . . . , Xn}. Does the sequence Z1, Z2, Z3, . . . converge in probabil-ity? If yes, to what? If not, just say “no” without explanation. Problem 2. (68 points) Alice shows up at an Athena cluster at time zero and spends her time exclusively in typing emails. The times that her emails are sent are a Poisson process with rate λA per hour. 1. (3 points) What is the probability that Alice sent exactly three emails during the time interval [1, 2]? 2. Let Y1 and Y2 be the times at which Alice’s first and second emails were sent. (a) (3 points) Find E[Y2 | Y1]. (b) (3 points) Find the PDF of Y12 . (c) (3 points) Find the joint PDF of Y1 and Y2. Page 3 of 20 **Athena is MIT's UNIX-based computing environment. OCW does not provide access to it.Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) 3. You show up at time 1 and you are told that Alice has sent exactly one email so far. (Only give answers here, no need to justify them.) (a) (3 points) What is the conditional expectation of Y2 given this information? (b) (3 points) What is the conditional expectation of Y1 given this information? 4. Bob just finished exercising (without email access) and sits next to Alice at time 1. He starts typing emails at time 1, and fires them according to an independent Poisson process with rate λB. (a) (5 points) What is the PMF of the total number of emails sent by the two of them together during the interval [0, 2]? (b) (5 points) What is the expected value of the total typing time associated with the email that Alice is typing at the time that Bob shows up? (Here, “total typing time” includes the time that Alice spent on that email both before and after Bob’s arrival.) (c) (5 points) What is the expected value of the time until each one of them has sent at least one email? (Note that we count time starting from time 0, and we take into account any emails possibly sent out by Alice during the interval [0, 1].) (d) (5 p oints) Given that a total of 10 emails were sent during the interval [0, 2], what is the probability that exactly 4 of them were sent by Alice? 5. (5 points) Suppose that λA = 4. Use Chebyshev’s inequality to find an upper bound on the probability that Alice sent at least 5 emails during the time interval [0, 1]. Does the Markov inequality provide a better bound? 6. (5 p oints) You do not know λA but you watch Alice for an hour and see that she sent exactly 5 emails. Derive the maximum likelihood estimate of λA based on this information. 7. (5 points) We have reasons to believe that λA is a large number. Let N be the number of emails sent during the interval [0, 1]. Justify why the CLT can be applied to N , and give a precise statement of the CLT in this case. 8. (5 points) Under the same assumption as in last part, that λA is large, you can now pretend that N is a normal random variable. Suppose that you observe the value of N. Give an (approx-imately) 95% confidence interval for λA. State precisely what approximations you are making. Possibly useful facts: The cumulative normal distribution satisfies Φ(1.645) = 0.95 and Φ(1.96) = 0.975. 9. You are now told that λA is actually the realized value of an exponential random variable Λ, with parameter 2: fΛ(λ) = 2e−2λ , λ ≥ 0. (a) (5 points) Find E[N2]. (b) (5 points) Find the linear least squares estimator of Λ given N. Page 4 of


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MIT 6 041 - Probabilistic Systems Analysis

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