6.041/6.431 Spring 2009 Final Exam Thursday, May 21, 1:30-4:30 PM.

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1 6.041/6.431 Sprig 2009 Fial Exam Thursday, May 21, 1:30-4:30 PM. Name: Recitatio Istructor: Questio Part Score Out of all 18 2 all 24 3 a 4 b 4 c 4 4 a 6 b 6 c 6 5 a 6 b 6 6 a 4 b 4 c 4 d 5 e 5 7 a 6 b 6 Total 120 Write your solutios i this quiz packet, oly solutios i the quiz packet will be graded. You are allowed three two-sided 8.5 by 11 formula sheet plus a calculator. You have 180 miutes to complete the quiz. Be eat! You will ot get credit if we ca t read it.

2 Massachusetts Istitute of Techology Departmet of Electrical Egieerig & Computer Sciece 6.041/6.431: Probabilistic Systems Aalysis (Sprig 2009) Problem 1: True or False (2pts. each, 18 pts. total) No partial credit will be give for idividual questios i this part of the quiz. a. Let {X } be a sequece of i.i.d radom variables takig values i the iterval [0,0.5]. Cosider the followig statemets: (A) If E[X 2 ] coverges to 0 as the X coverges to 0 i probability. (B) If all X have E[X ] = 0.2 ad var (X ) coverges to 0 as the X coverges to 0.2 i probability. (C) The sequece of radom variables Z, defied by Z = X 1 X 2 X, coverges to 0 i probability as. Which of these statemets are always true? Write True or False i each of the boxes below. A: B: C: b. Let X i (i = 1,2,... ) be i.i.d. radom variables with mea 0 ad variace 2; Y i (i = 1,2,... ) be i.i.d. radom variables with mea 2. Assume that all variables X i, Y j are idepedet. Cosider the followig statemets: (A) (B) 1 (C) X 1 + +X coverges to 0 i probability as. X 2 + +X 2 X 1 Y 1 + coverges to 2 i probability as. +X Y coverges to 0 i probability as. Which of these statemets are always true? Write True or False i each of the boxes below. A: B: C: c. We have i.i.d. radom variables X 1...X with a ukow distributio, ad with µ = E[X i ]. We defie M = (X X )/. Cosider the followig statemets: (A) M is a maximum-likelihood estimator for µ, irrespective of the distributio of the X i s. (B) M is a cosistet estimator for µ, irrespective of the distributio of the X i s. (C) M is a asymptotically ubiased estimator for µ, irrespective of the distributio of the X i s. Which of these statemets are always true? Write True or False i each of the boxes below. A: B: C: Page 3 of 15

3 Massachusetts Istitute of Techology Departmet of Electrical Egieerig & Computer Sciece 6.041/6.431: Probabilistic Systems Aalysis (Sprig 2009) Problem 2: Multiple Choice (4 pts. each, 24 pts. total) Clearly circle the appropriate choice. No partial credit will be give for idividual questios i this part of the quiz. a. Earthquakes i Sumatra occur accordig to a Poisso process of rate λ = 2/year. Coditioed o the evet that exactly two earthquakes take place i a year, what is the probability that both earthquakes occur i the first three moths of the year? (for simplicity, assume all moths have 30 days, ad each year has 12 moths, i.e., 360 days). (i) 1/12 (ii) 1/16 (iii) 64/225 (iv) 4e 4 (v) There is ot eough iformatio to determie the required probability. (vi) Noe of the above. b. Cosider a cotiuous-time Markov chai with three states i {1,2,3}, with dwellig time i each visit to state i beig a expoetial radom variable with parameter ν i = i, ad trasitio probabilities p ij defied by the graph What is the log-term expected fractio of time spet i state 2? (i) 1/2 (ii) 1/4 (iii) 2/5 (iv) 3/7 (v) Noe of the above. Page 4 of 15

4 Massachusetts Istitute of Techology Departmet of Electrical Egieerig & Computer Sciece 6.041/6.431: Probabilistic Systems Aalysis (Sprig 2009) c. Cosider the followig Markov chai: Startig i state 3, what is the steady-state probability of beig i state 1? (i) 1/3 (ii) 1/4 (iii) 1 (iv) 0 (v) Noe of the above. d. Radom variables X ad Y are such that the pair (X,Y ) is uiformly distributed over the trapezoid A with corers (0,0), (1,2), (3,2), ad (4,0) show i Fig. 1: X Y Figure 1: f X,Y (x,y) is costat over the shaded area, zero otherwise. i.e. { c, f X,Y (x,y) = 0, (x,y) A else. We observe Y ad use it to estimate X. Let Xˆ be the least mea squared error estimator of X give Y. What is the value of var(xˆ X Y = 1)? (i) 1/6 (ii) 3/2 (iii) 1/3 (iv) The iformatio is ot sufficiet to compute this value. (v) Noe of the above. Page 5 of 15

5 (i) [M δ (ii) [M δ (iii) [M δ (iv) [M δ Massachusetts Istitute of Techology Departmet of Electrical Egieerig & Computer Sciece 6.041/6.431: Probabilistic Systems Aalysis (Sprig 2009) e. X 1...X are i.i.d. ormal radom variables with mea value µ ad variace v. Both µ ad v are ukow. We defie M = (X X )/ ad,m + δ,m + δ,m + δ,m + δ 1 V = (X i M ) 2 1 i=1 We also defie Φ(x) to be the CDF for the stadard ormal distributio, ad Ψ 1 (x) to be the CDF for the t-distributio with 1 degrees of freedom. Which of the followig choices gives a exact 99% cofidece iterval for µ for all > 1? V V V V (v) Noe of the above. V ] where δ is chose to give Φ(δ) = V ] where δ is chose to give Φ(δ) = V ] where δ is chose to give Ψ 1 (δ) = V ] where δ is chose to give Ψ 1 (δ) = f. We have i.i.d. radom variables X 1,X 2 which have a expoetial distributio with ukow parameter θ. Uder hypothesis H 0, θ = 1. Uder hypothesis H 1, θ = 2. Uder a likelihood-ratio test, the rejectio regio takes which of the followig forms? (i) R = {(x 1,x 2 ) : x 1 + x 2 > ξ} for some value ξ. (ii) R = {(x 1,x 2 ) : x 1 + x 2 < ξ} for some value ξ. (iii) R = {(x 1,x 2 ) : e x 1 + e x 2 > ξ} for some value ξ. (iv) R = {(x 1,x 2 ) : e x 1 + e x 2 < ξ} for some value ξ. (v) Noe of the above. Page 6 of 15

6 Problem 3 (12 pts. total) Massachusetts Istitute of Techology Departmet of Electrical Egieerig & Computer Sciece 6.041/6.431: Probabilistic Systems Aalysis (Sprig 2009) Alies of two races (blue ad gree) are arrivig o Earth idepedetly accordig to Poisso process distributios with parameters λ b ad λ a respectively. The Alie Arrival Registratio Service Authority (AARSA) will begi registerig alie arrivals soo. Let T 1 deote the time AARSA will fuctio util it registers its first alie. Let G be the evet that the first alie to be registered is a gree oe. Let T 2 be the time AARSA will fuctio util at least oe alie of both races is registered. (a) (4 poits.) Express µ 1 = E[T 1 ] i terms of λ g ad λ b. Show your work. (b) (4 poits.) Express p = P(G) i terms of λ g ad λ b. Show your work. Page 7 of 15

7 Massachusetts Istitute of Techology Departmet of Electrical Egieerig & Computer Sciece 6.041/6.431: Probabilistic Systems Aalysis (Sprig 2009) (c) (4 poits.) Express µ 2 = E[T 2 ] i terms of λ g ad λ b. Show your work. Page 8 of 15

8 Problem 4 (18 pts. total) Massachusetts Istitute of Techology Departmet of Electrical Egieerig & Computer Sciece 6.041/6.431: Probabilistic Systems Aalysis (Sprig 2009) Researcher Jill is iterested i studyig employmet i techology firms i Dilico Valley. She deotes by X i the umber of employees i techology firm i ad assumes that X i are idepedet ad idetically distributed with mea p. To estimate p, Jill radomly iterviews techology firms ad observes the umber of employees i these firms. (a) (6 poits.) Jill uses X X M = as a estimator for p. Fid the limit of P(M x) as for x < p. Fid the limit of P(M x) as for x > p. Show your work. (b) (6 poits.) Fid the smallest, the umber of techology firms Jill must sample, for which the Chebyshev iequality yields a guaratee P( M p 0.5) Assume that var (X i ) = v for some costat v. State your solutio as a fuctio of v. Show your work. Page 9 of 15

9 Massachusetts Istitute of Techology Departmet of Electrical Egieerig & Computer Sciece 6.041/6.431: Probabilistic Systems Aalysis (Sprig 2009) (c) (6 poits.) Assume ow that the researcher samples = 5000 firms. Fid a approximate value for the probability P( M 5000 p 0.5) usig the Cetral Limit Theorem. Assume agai that var (X i ) = v for some costat v. Give your aswer i terms of v, ad the stadard ormal CDF Φ. Show your work. Page 10 of 15

10 Problem 5 (12 pts. total) Massachusetts Istitute of Techology Departmet of Electrical Egieerig & Computer Sciece 6.041/6.431: Probabilistic Systems Aalysis (Sprig 2009) The RadomView widow factory produces widow paes. After maufacturig, 1000 paes were loaded oto a truck. The weight W i of the i-th pae (i pouds) o the truck is modeled as a radom variable, with the assumptio that the W i s are idepedet ad idetically distributed. (a) (6 poits.) Assume that the measured weight of the load o the truck was 2340 pouds, ad that var (W i ) 4. Fid a approximate 95 percet cofidece iterval for µ = E[W i ], usig the Cetral Limit Theorem (you may use the stadard ormal table which was haded out with this quiz). Show your work. (b) (6 poits.) Now assume istead that the radom variables W i are i.i.d, with a expoetial distributio with parameter θ > 0, i.e., a distributio with PDF f W (w;θ) = θe θw What is the maximum likelihood estimate of θ, give that the truckload has weight 2340 pouds? Show your work. Page 11 of 15

11 Problem 6 (21 pts. total) Massachusetts Istitute of Techology Departmet of Electrical Egieerig & Computer Sciece 6.041/6.431: Probabilistic Systems Aalysis (Sprig 2009) I Alice s Woderlad, there are six differet seasos: Fall (F), Witer (W), Sprig (Sp), Summer (Su), Bitter Cold (B), ad Golde Sushie (G). The seasos do ot follow ay particular order, istead, at the begiig of each day the Head Wizard assigs the seaso for the day, accordig to the followig Markov chai model: Thus, for example, if it is Fall oe day the there is 1/6 probability that it will be Witer the ext day (ote that it is possible to have the same seaso agai the ext day). (a) (4 poits.) For each state i the above chai, idetify whether it is recurret or trasiet. Show your work. (b) (4 poits.) If it is Fall o Moday, what is the probability that it will be Summer o Thursday of the same week? Show your work. Page 12 of 15

12 Massachusetts Istitute of Techology Departmet of Electrical Egieerig & Computer Sciece 6.041/6.431: Probabilistic Systems Aalysis (Sprig 2009) (c) (4 poits.) If it is Sprig today, will the chai coverge to steady-state probabilities? If so, compute the steady-state probability for each state. If ot, explai why these probabilities do ot exist. Show your work. (d) (5 poits.) If it is Fall today, what is the probability that Bitter Cold will ever arrive i the future? Show your work. Page 13 of 15

13 Massachusetts Istitute of Techology Departmet of Electrical Egieerig & Computer Sciece 6.041/6.431: Probabilistic Systems Aalysis (Sprig 2009) (e) (5 poits.) If it is Fall today, what is the expected umber of days till either Summer or Golde Sushie arrives for the first time? Show your work. Page 14 of 15

14 Problem 7 (12 pts. total) Massachusetts Istitute of Techology Departmet of Electrical Egieerig & Computer Sciece 6.041/6.431: Probabilistic Systems Aalysis (Sprig 2009) A ewscast coverig the fial baseball game betwee Sed Rox ad Y Nakee becomes oisy at the crucial momet whe the viewers are iformed whether Y Nakee wo the game. Let a be the parameter describig the actual outcome: a = 1 if Y Nakee wo, a = 1 otherwise. There were viewers listeig to the telecast. Let Y i be the iformatio received by viewer i (1 i ). Uder the oisy telecast, Y i = a with probability p, ad Y i = a with probability 1 p. Assume that the radom variables Y i are idepedet of each other. The viewers as a group come up with a joit estimator { 1 if i=1 Y i 0, Z = 1 otherwise. (a) (6 poits.) Fid lim P(Z = a) assumig that p > 0.5 ad a = 1. Show your work. (b) (6 poits.) Fid lim P(Z = a), assumig that p = 0.5 ad a = 1. Show your work. Page 15 of 15

15 MIT OpeCourseWare / Probabilistic Systems Aalysis ad Applied Probability Fall 2010 For iformatio about citig these materials or our Terms of Use, visit:

This exam contains 19 pages (including this cover page) and 10 questions. A Formulae sheet is provided with the exam.

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