6.041/6.431 Spring 2009 Final ExamThursday, May 21, 1:30 - 4:30 PM.Name:Recitation Instructor:Question Part Score Out of0 21 all 182 all 243 a 4b 4c 44 a 6b 6c 65 a 6b 66 a 4b 4c 4d 5e 57 a 6b 6Total 120• Write your solutions in this quiz packet, only solutions in the quiz packet will be graded.• You are allowed three two-sided 8.5 by 11 formula sheet plus a calculator.• You have 180 minutes to complete the quiz.• Be neat! You will not get credit if we can’t read it.Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Spring 2009)Problem 0 (2 pts.)Write your name, and your assignedrecitation instructor’s name, on the cover of th e quiz booklet.The In s tr uctors are listed below.Recitation Instructor Recitation TimeDevavrat Shah 10 & 11 AMShivani Agarwal 11AM & 12PMAsu Ozdaglar 12 & 1 PMPablo Parrilo (6.431) 10 & 11AMPage 2 of 15Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Spring 2009)Problem 1: True or False (2pts. each, 18 pts. total)No partial credit will be given for individual questions in this p art of the quiz.a. Let {Xn} be a sequence of i.i.d random variables taking values in the interval [0, 0.5]. Considerthe following statements:(A) If E[X2n] converges to 0 as n → ∞ then Xnconverges to 0 in probability.(B) If all Xnhave E[Xn] = 0.2 and var (Xn) converges to 0 as n → ∞ then Xnconverges to 0.2in pr obab ility.(C) The sequence of random variables Zn, defined by Zn= X1· X2· · · Xn, converges to 0 inprobability as n → ∞.Which of these statements are always true? Write True or False in each of the boxes below.A: B: C:b. Let Xi(i = 1, 2, . . . ) be i.i.d. r andom variables with mean 0 and variance 2; Yi(i = 1, 2, . . . ) bei.i.d. random variables with mean 2. Assume th at all variables Xi, Yjare in dependent. Considerthe following statements:(A)X1+···+Xnnconverges to 0 in probability as n → ∞.(B)X21+···+X2nnconverges to 2 in probability as n → ∞.(C)X1Y1+···+XnYnnconverges to 0 in probability as n → ∞.Which of these statements are always true? Write True or False in each of the boxes below.A: B: C:c. We have i.i.d. random variables X1. . . Xnwith an unknown distribution, and with µ = E[Xi].We define Mn= (X1+ . . . + Xn)/n. Consider the following statements:(A) Mnis a maximum-likelihood estimator for µ, irrespective of the distribution of th e Xi’s.(B) Mnis a consistent estimator for µ, irrespective of the distribution of the Xi’s.(C) Mnis an asymptotically unbiased estimator for µ, irrespective of the distribution of theXi’s.Which of these statements are always true? Write True or False in each of the boxes below.A: B: C:Page 3 of 15Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Spring 2009)Problem 2: Multiple Choice (4 pts. each, 24 pts. total)Clearly circle the appropriate choice. No partial credit w ill be given for individual questions in thispart of the quiz.a. Earthquakes in Sumatra occur according to a Poisson process of rate λ = 2/year. Conditionedon the event that exactly two earthquakes take place in a year, what is the probability that bothearthquakes occur in the first three months of the year? (for simplicity, assume all months have30 days, and each year has 12 months, i.e., 360 days).(i) 1/12(ii) 1/16(iii) 64/225(iv) 4e−4(v) There is not enough information to determine the required probability.(vi) None of the above.b. Consider a continu ous-time Markov chain with three states i ∈ {1, 2, 3}, with dwelling time ineach visit to state i being an exponential random variable with parameter νi= i, and transitionprobabilities pijdefined by the graphWhat is the long-term expected fraction of time spent in state 2?(i) 1/2(ii) 1/4(iii) 2/5(iv) 3/7(v) None of the above.Page 4 of 15Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Spring 2009)c. Consider th e following Markov chain:Starting in state 3, what is the steady-state probability of being in state 1?(i) 1/3(ii) 1/4(iii) 1(iv) 0(v) None of the above.d. Random variables X and Y are such that the pair (X, Y ) is uniformly distributed over thetrapezoid A with corners (0, 0), (1, 2), (3, 2), and (4, 0) shown in Fig. 1:4YX231Figure 1: fX,Y(x, y) is constant over the shaded area, zero otherwise.i.e.fX,Y(x, y) =(c , (x, y) ∈ A0 , else .We observe Y and use it to estimate X. LetˆX be the least mean squared error estimator of Xgiven Y . Wh at is the value of var(ˆX − X|Y = 1)?(i) 1/6(ii) 3/2(iii) 1/3(iv) The information is not suffi cient to compute this value.(v) None of the above.Page 5 of 15Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Spring 2009)e. X1. . . Xnare i.i.d. normal random variables with mean value µ an d variance v. Both µ and vare unknown. We define Mn= (X1+ . . . + Xn)/n andVn=1n − 1nXi=1(Xi− Mn)2We also define Φ(x) to be the CDF for the s tandard normal distribution, and Ψn−1(x) to be theCDF for the t-distribution with n − 1 degrees of freedom. Which of the following choices givesan exact 99% confidence interval for µ for all n > 1?(i) [Mn− δqVnn, Mn+ δqVnn] where δ is chosen to give Φ(δ) = 0.99.(ii) [Mn− δqVnn, Mn+ δqVnn] where δ is chosen to give Φ(δ) = 0.995.(iii) [Mn− δqVnn, Mn+ δqVnn] where δ is chosen to give Ψn−1(δ) = 0.99.(iv) [Mn− δqVnn, Mn+ δqVnn] where δ is chosen to give Ψn−1(δ) = 0.995.(v) None of the above.f. We have i.i.d. random variables X1, X2which have an exponential distribution with unknownparameter θ. Under hypothesis H0, θ = 1. Under hypothesis H1, θ = 2. Under a likelihood-ratiotest, the rejection region takes which of the following forms?(i) R = {(x1, x2) : x1+ x2> ξ} for some value ξ.(ii) R = {(x1, x2) : x1+ x2< ξ} for some value ξ.(iii) R = {(x1, x2) : ex1+ ex2> ξ} f or some value ξ.(iv) R = {(x1, x2) : ex1+ ex2< ξ} f or some value ξ.(v) None of the above.Page 6 of 15Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Spring 2009)Problem 3 (12 pts. total)Aliens of two races (blue and green) are arriving on Earth independently according to
View Full Document
Unlocking...