6.041/6.431 Spring 2009 Quiz 1Wednesday, March 11, 7:30 - 9:30 PM.Name:Recitation Instructor:TA:Question Part Score Out of0 31 all 402 a 5b 5c 6d 63 a 5b 6c 6d 6e 6f 6g 106.041 Total 1006.431 Total 110• Write your solutions in this quiz packet, only solutions in the quiz packet will be graded.• Question one, multiple choice questions, will receive no partial credit. Partial credit forquestion two and three will be awarded.• You are allowed one two-sided 8.5 by 11 formula sheet plus a calculator.• You have 120 minutes to complete the quiz.• Be neat! You will not get credit if we can’t read it.• Graded quizzes will be returned in recitation on Tuesday 3/17.Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Spring 2009)Problem 0: (3 pts) Write your name, your assigned recitation instructor’s name, and assigned TA’sname on the cover of the quiz booklet. The Instructor/TA pairing is listed below.Recitation Instructor TA Recitation TimeVivek Goyal Natasa Blitvic 10 & 11 AMMichael Collins Danielle Hinton 10 & 11 AMShivani Agarwal Stavros Valavanis 12 & 1 PMDimitri Bertsekas (6.431) Aman Chawla (6.431) 1 & 2 PMPage 2 of 11Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Spring 2009)Question 1Multiple Choice Questions: CLEARLY circle the appropriate choice. Scratch paper is available ifneeded, though NO partial credit will be given for the Multiple Choice.a. Which of the following statements is NOT true?(i) If A ⊂ B, then P(A) ≤ P(B).(ii) If P(B) > 0, then P(A|B) ≥ P(A).(iii) P(A ∩ B) ≥ P(A) + P(B) − 1.(iv) P(A ∩ Bc) = P(A ∪ B) − P(B).b. We throw n identical balls into m urns at random, where each urn is equally likely and eachthrow is independent of any other throw. What is the probability that the i-th urn is empty?(i)1 −1mn(ii)1 −1nm(iii)mn1 −1nm(iv)nm1mnc. We toss two fair coins simultaneously and independently. If the outcomes of the two coins arethe same, we win; otherwise, we lose. Let A be the event that the first coin comes up heads, Bbe the event that the second coin comes up heads, and C be the event that we win. Which ofthe following statements is false?(i) Events A and B are independent.(ii) Events A and C are not independent.(iii) Events A and B are not conditionally independent given C.(iv) The probability of winning is 1/2.d. For a biased coin, the probability of “heads” is 1/3. Let h be the number of heads in fiveindependent coin tosses. What is the probability P(first toss is a head | h = 1 or h = 5)?(i)13(23)4513(23)4+(13)5(ii)13(23)413(23)4+(13)5(iii)13(23)4+(13)5513(23)4+(13)5(iv)15Page 3 of 11Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Spring 2009)e. A well-shuffled deck of 52 cards is dealt evenly to two players (26 cards each). What is theprobability that player 1 gets all the aces?(i)0@48221A0@52261A(ii)40@48221A0@52261A(iii)48!22!52!26!(iv)4!0@48221A0@52261Af. Suppose X, Y and Z are three independent discrete random variables. Then, X and Y + Z are(i) always(ii) sometimes(iii) neverindependent.g. To obtain a driving licence, Mina needs to pass her driving test. Every time Mina takes a drivingtest, with probability 1/2, she will clear the test independent of her past. Mina failed her firsttest. Given this, let Y be the additional number of tests Mina takes before obtaining a licence.Then,(i) E[Y ] = 1.(ii) E[Y ] = 2.(iii) E[Y ] = 0.h. Consider two random variables X and Y , each taking values in {1, 2, 3}. Let their joint PMF besuch that for any 1 ≤ x, y ≤ 3,PX,Y(x, y) =(0 if (x, y) ∈ {(1, 3), (2, 1), (3, 2)}strictly positive otherwise.Then,(i) X and Y can be independent or dependent depending upon the strictly positive values.(ii) X and Y are always independent.(iii) X and Y can never be independent.Page 4 of 11Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Spring 2009)i. Suppose you play a matching coins game with your friend as follows. Both you and your friendhave a coin. Each time, you two reveal a side (i.e. H or T) of your coin to each other simulta-neously. If the sides match, you WIN a 1 from your friend and if sides do not match then youlose a 1 to your friend. Your friend has a complicated (unknown) strategy in selecting the sidesover time. You decide to go with the following simple strategy. Every time, you will toss yourunbiased coin independently of everything else, and you will reveal its outcome to your friend (ofcourse, your friend does not know the outcome of your random toss until you reveal it). Then,(i) On average, you will lose money to your smart friend.(ii) On average, you will neither lose nor win. That is, your average gain/loss is 0.(iii) On average, you will make money from your friend.j. Let Xi, 1 ≤ i ≤ 4 be independent Bernoulli random variable each with mean p = 0.1. LetX =P4i=1Xi. That is, X is a Binomial random variable with parameters n = 4 and p = 0.1.Then,(i) E[X1|X = 2] = 0.1.(ii) E[X1|X = 2] = 0.5.(iii) E[X1|X = 2] = 0.25.Page 5 of 11Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Spring 2009)Question 2:Alice and Bob both need to buy a bicycle. The bike store has a stock of four green, three yellow,and two red bikes. Alice randomly picks one of the bikes and buys it. Immediately after, Bob doesthe same. The sale price of the green, yellow, and red bikes are $300, $200 and $100, respectively.Let A be the event that Alice bought a green bike, and B be the event that Bob bought a greenbike.a. What is P(A)? What is P(A|B)?b. Are A and B independent events? Justify your answer.c. What is the probability that at least one of them bought a green bike?Page 6 of 11Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Spring 2009)d. What is the probability that Alice and Bob bought bicycles of different colors?e. Given that Bob bought a green bike, what is the expected value of the amount of money spentby Alice?f. Let G be the number of green bikes that remain on the store after Alice and Bob’s visit. ComputeP(B|G = 3).Page 7 of
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