Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2008)Problem Set 8Due: November 5, 2008Topics: Markov and Chebyshev bounds, weak law of large numbers, central limittheorem, Bernoulli random processes [Text sections: 5.1-5.4 & 6.1]1. Joe the Plumber has been having trouble keeping the income at his plumbing business steadylately. He notices that his pr ofit each week is uniformly distributed between $1,800 and$8,600. Being a small business owner, he doesnt have the stomach for these fluctuations,especially because they introduce additional uncertainty as to whether or not he’ll soon facehigher taxes. Assume th at each week’s profit is independent of the profits of all other weeks.Find a go od app roximation to the probability that over the course of a year (52 weeks), Joe’splumbing business will have a yearly income of less than $250,000.2. On any given flight, an airline tries to sell as many tickets as possible. Sup pose, on average,10% of ticket holders fail to show up, all independent of each other. Knowing this, anairline will sell more tickets than available seats (overbook the flight) and hope that thereare sufficient numbers of passengers who do not show up to compensate for its overbooking.Using the central limit theorem, determine n, the maximum number tickets an airline shouldsell on a flight with 300 seats so that it can be approximately 99% confident that it need notdeny boarding to any of the n passengers holding tickets.3. Parking your car at MIT is expensive, so expensive that you decide that it must be cheaperto keep your car parked illegally in a tow zone and chance the occasional ticket or tow. Oneach d ay, a Cambridge meter maid will notice your illegally parked car with probability 0.25.Upon noticing your illegally parked car, with probability 0.8 he or she will only issue you aticket; otherwise your car will be towed. All of this occurs independently on each day, andindependent of what happens on other days.(a) What is the expected time (in days) between successive times your car is towed?(b) What is the standard deviation of the time (in days) between successive times your caris towed?(c) What is the expected number of times your car is towed over the cours e of two months(60 days)?Suppose that in addition to the city of Cambridge, your car may also be towed by MIT.On each day with probability 0.1, an MIT tow truck will pass by your ‘parking space,’independently of all other days. If you car has not yet been towed by the city of Cambridge,the MIT tow truck w ill tow away your car. Assume that MIT operates independently ofCambridge’s meter maids.(d) What is the PMF of the time (in days) till your car is towed a seventh time?4. Tina Fey goes trick-or-treating the night of Halloween dressed as Sarah Palin. S he visitshomes in her neighborhood to collect candy, but only receives cand y, naturally, when thedoor is an s wered and the family still has a piece of candy to give away. Upon knocking,the probability of the door being answered is 3/4, and the probability that the home stillPublished October 31, 2008 Page 1 of 4Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2008)has candy is 2/3. Assume that the events “Door answered” and “candy remaining” areindependent and also that the outcomes at each home are independent. Also assume thateach home gives away at most a single piece of candy.(a) Determine the probability that Tina receives her first piece of candy at the third hous eshe visits.(b) Given th at she has received exactly four pieces of candy from the first eight houses,determine the conditional probability that Tina w ill receive her fifth piece of candy atthe eleventh house.(c) Determine the probability th at she receives her second piece of candy at the fifth house.(d) Given that she did not receive her second piece of candy at the second house, determinethe conditional probability that she will receive her second piece of candy at the fif thhouse.(e) We will say that Tina “needs a new bag” immediately after the house from which shereceives a piece of candy filling her bag to the top. If she starts out with a bag that hasroom for only two pieces of candy, determine the probability that she visits at least fivehomes before needing a new bag.(f) If she starts out with a bag with space for m pieces of cand y, determine the expectedvalue and variance of Cm, the number of homes with candy she passes up (because ofno answer) before she needs a new bag.5. This problem deals with the long-term outcome of multiplicative gambling or investmentmodels. In particular, we will focus on the long-term outcome of the ‘Double or Quarter’game wh ich was considered in Lecture 1 and on the first quiz, and which is a good example ofsuch multiplicative models. Recall that in the ‘Double or Quarter’ game, a fair coin is tossedrepeatedly; each time it comes up heads, our investment is doubled, and each time it comesup tails,34of our investment is lost. Thus, if Xkis the return on the kth trial, i.e., the ratioby which our investment is multiplied on the kth round, thenP (Xk= 2) = PXk=14=12.Assume we initially invest $1. If we reinvest all of our holdings on each subsequent trial, ourwealth Wn(in dollars) after n ≥ 1 trials is given byWn= X1· X2· . . . · Xn.You showed on th e quiz thatE[Wn] =98n,which implies thatE[Wn] → ∞ as n → ∞ . (1)In other words, our expected wealth after n rounds converges to infinity as n → ∞. Does thismean th at if we continue investing u nder this model for a large number of rounds n, then weare likely to end up with a large wealth Wn? Let us look at this question more carefully.Published October 31, 2008 Page 2 of 4Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2008)(a) Show that the probability of making a profit after n rounds, P(Wn> 1), converges tozero as n → ∞. Why does this not contradict our earlier finding in Equation (1)?Hint: Note that Wnis greater than 1 if and only if the number of heads in the n tosses,Hn, is more th an twice th e number of tails, n − Hn. Thus P(Wn> 1) = P(Hn>2(n − Hn)) = P(Hn>2n3). Use Chebyshev’s inequality to obtain an upper bound onthis probability, and show that this
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