6.041/6.431 Fall 2010 Quiz 1Tuesday, October 12, 7:30 - 9:00 PM.DO NOT TURN THIS PAGE OVER UNTILYOU ARE TOLD TO DO SOName:Recitation Instructor:TA:Question Score Out of1.1 101.2 101.3 101.4 101.5 51.6 101.7 101.8 102.1 102.2 102.3 10Your Grade 105• This quiz has 2 problems, worth a total of 105 points.• You may tear apart pages 3, 4 and 5, as per your convenience, but you must turn themin 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 one two-sided, handwritten, 8.5 by 11 formula sheet. Calculators are notallowed.• You may give an answer in the form of an arithmetic expression (sums, prod-ucts, ratios, factorials) of numbers that could be evaluated using a calculator.Expressions like83orP5k=0(1/2)kare also fine.• You have 90 minutes to complete the quiz.• Graded quizzes will be returned in recitation on Thursday 10/14.Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2010)Problem 0: (0 points) Write your name, your assigned recitation instructor’s name, and assignedTA’s name on the cover of the quiz booklet. The Instructor/TA pairing is listed below.Recitation Instructor TA Recitation TimeVivek Goyal Uzoma Orji 10 & 11 AMPeter Hagelstein Ahmad Zamanian 12 & 1 PMAli Shoeb Shashank Dwivedi 2 PMDimitri Bertsekas (6.431) Aliaa Atwi 2 & 3 PMPage 2 of 13Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2010)116 Discrete Random Variables Chap. 2of a discrete random variable. We showed how to use the PMF of a randomvariable X to calculate the mean and the variance of a related random variableY = g(X) without calculating the PMF of Y . In the special case where g isa linear function, Y = aX + b, the means and the variances of X and Y arerelated byE[Y ]=aE[X]+b, var(Y )=a2var(X).We also discussed several special random variables, and derived their PMF,mean, and variance, as summarized in the table that follows.Summary of Results for Special Random VariablesDiscrete Uniform over [a, b]:pX(k)=!1b − a +1, if k = a, a +1,...,b,0, otherwise,E[X]=a + b2, var(X)=(b − a)(b − a + 2)12.Bernoulli with Parameter p: (Describes the success or failure in a singletrial.)pX(k)="p, if k = 1,1 − p, if k = 0,E[X]=p, var(X)=p(1 − p).Binomial with Parameters p and n: (Describes the number of successesin n independent Bernoulli trials.)pX(k)=#nk$pk(1 − p)n−k,k=0, 1, . . . , n,E[X]=np, var(X)=np(1 − p).Geometric with Parameter p: (Describes the number of trials until thefirst success, in a sequence of independent Bernoulli trials.)pX(k) = (1 − p)k−1p, k =1, 2,...,E[X]=1p, var(X)=1 − pp2.Page 3 of 13Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2010)Problem 1: (75 points)Note: All parts can be done independently, with the exception of the last part. Just in case youmade a mistake in the previous part, you can use a symbol for the expression you found there, anduse that symbol in the formulas for the last part.Note: Algebraic or numerical expressions do not need to be simplified in your answers.Jon and Stephen cannot help but think about their commutes using probabilistic modeling. Bothof the them start promptly at 8am.Stephen drives and thus is at the mercy of traffic lights. When all traffic lights on his route aregreen, the entire trip takes 18 minutes. Stephen’s route includes 5 traffic lights, each of which is redwith probability 1/3, independent of every other light. Each red traffic light that he encounters adds1 minute to his commute (for slowing, stopping, and returning to speed).1. (10 points) Find the PMF, expectation, and variance of the length (in minutes) of Stephen’scommute.2. (10 points) Given that Stephen’s commute took him at most 19 minutes, what is the expectednumber of red lights that he encountered?3. (10 points) Given that the last red light encountered by Stephen was the fourth light, what isthe conditional variance of the total number of red lights he encountered?4. (10 points) Given that Stephen encountered a total of three red lights, what is the probabilitythat exactly two out of the first three lights were red?Jon’s commuting behavior is rather simple to model. Jon walks a total of 20 minutes from hishome to a station and from a station to his office. He also waits for X minutes for a subway train,where X has the discrete uniform distribution on {0, 1, 2, 3}. (All four values are equally likely, andindependent of the traffic lights encountered by Stephen.)5. (5 points) What is the PMF of the length of Jon’s commute in minutes?6. (10 points) Given that there was exactly one person arriving at exactly 8:20am, what is theprobability that this person was Jon?7. (10 points) What is the probability that Stephen’s commute takes at most as long as Jon’scommute?8. (10 points) Given that Stephen’s commute took at most as long as Jon’s, what is the conditionalprobability that Jon waited 3 minutes for his train?Problem 2. (30 points) For each one of the statements below, give either a proof or a counterexampleshowing that the statement is not always true.1. (10 points) If events A and B are independent, then the events A and Bcare also independent.2. (10 points) Let A, B, and C be events associated with a common probabilistic model, andassume that 0 < P(C) < 1. Suppose that A and B are conditionally independent given C.Then, A and B are conditionally independent given Cc.Page 4 of 13Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2010)3. (10 points) Let X and Y be independent random variables. Then, var(X + Y ) ≥ var(X).Each question is repeated in the following pages. Please write your answer onthe appropriate page.Page 5 of 13Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2010)Problem 1: (75 points)Note: All parts can be done independently, with the exception of the last part. Just in case youmade a mistake in the previous part, you can use a symbol for the expression you found there, anduse that symbol in the
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