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# I N T R O D U C T I O N T O P R O B A B I L I T Y …

I N T **R O D** U C T I **O** N T **O** P **R O B** A **B** I L I T Y by Dimitri P. Bertsekas and John N. Tsitsiklis C H A P T E **R** 2 : A **D D** I T I **O** N A L P **R O B** L E M S

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I N T R O D U C T I O N T O P R O B A B I L I T YbyDimitri P. Bertsekas and John N. TsitsiklisC H A P T E R 2 : A D D I T I O N A L P R O B L E M SSECTION Probability Mass FunctionsProblem probability of a royal flush in poker isp= 1/649,740. Show thatapproximately 649,740 hands would have to be dealt in order that the probability ofgetting at least one royal flush is above 1 1 annual premium of a special kind of insurance starts at $1000 andis reduced by 10% after each year where no claim has been filed. The probability thata claim is filed in a given year is , independently of preceding years. What is thePMF of the total premium paid up to and including the year when the first claim isfiled?SECTION Functions of Random VariablesProblem a discrete random variable that is uniformly distributed overthe set of integers in the range [a,b], whereaandbare integers witha <0< b. Findthe PMF of the random variables max{0,X}and min{0,X}.Problem a discrete random variable, and letY=|X|.(a) Assume that the PMF ofXispX(x) ={Kx2ifx= 3, 2, 1,0,1,2,3,0otherwise,whereKis a suitable constant. Determine the value ofK.(b) For the PMF ofXgiven in part (a) calculate the PMF ofY.(c) Give a general formula for the PMF ofYin terms of the PMF Expectation, Mean, and VarianceProblem a random variable that takes integer values and is symmetric,that is,P(X=k) =P(X= k) for all integersk. What is the expected value ofY= cos(X ) andY= sin(X )?Problem are visiting the rainforrest, but unfortunately your insect repellenthas run out. As a result, at each second, a mosquito lands on your neck with If a mosquito lands, it will bite you with probability , and it will never botheryou with probability , independently of other mosquitoes. What is the expectedtime between successive bites?Problem and Spassky play a sudden-death chess match whereby the firstplayer to win a game wins the match. Each game is won by Fischer with probabilityp, by Spassky with probabilityq, and is a draw with probability 1 p q.(a) What is the probability that Fischer wins the match?(b) What is the PMF, the mean, and the variance of the duration of the match?Problem particular binary data transmission and reception device is prone tosome error when receiving data. Suppose that each bit is read correctly with probabilityp. Find a value ofpsuch that when 10,000 bits are received, the expected number oferrors is at most a TV game show where each contestantispins an infinitelycalibrated wheel of fortune, which assigns him/her with some real number with avalue between 1 and 100. All values are equally likely and the value obtained by eachcontestant is independent of the value obtained by any other contestant.(a) FindP(X1< X2).(b) FindP(X1< X2,X1< X3).(c) LetNbe the integer-valued random variable whose value is the index of the firstcontestant who is assigned a smaller number than contestant 1. As an illustration,if contestant 1 obtains a smaller value than contestants 2 and 3 but contestant 4has a smaller value than contestant 1 (X4< X1), thenN= 4. FindP(N > n)as a function ofn.(d) FindE[N], assuming an infinite number of a nonnegative integer-valued random variable. Show thatE[N] = i=1P(N i).Problem ,...,Xnbe independent, identically distributed random vari-ables with common mean and variance. Find the values ofcanddthat will make thefollowing formula true:E[(X1+ +Xn)2]=cE[X1]2+d(E[X1]) Joint PMFs of Multiple Random VariablesProblem MIT football team wins any one game with probabilityp, andloses it with probability 1 p. Its performance in each game is independent of itsperformance in other games. LetL1be the number of losses before its first win, andletL2be the number of losses after its first win and before its second win. Find thejoint PMF class ofnstudents takes a test in which each student gets an A withprobabilityp, a B with probabilityq, and a grade below B with probability 1 p q,independently of any other student. IfXandYare the numbers of students that getan A and a B, respectively, calculate the joint PMFpX, ConditioningProblem probability class has 250 undergraduate students and 50 graduatestudents. The probability of an undergraduate (or graduate) student getting an A is1/3 (or 1/2, respectively). LetXbe the number of students that get an A in yourclass.(a) CalculateE[X] by first finding the PMF ofX.(b) CalculateE[X] by viewingXas a sum of random variables, whose mean is scalper is considering buying tickets for a particular game. Theprice of the tickets is $75, and the scalper will sell them at $150. However, if she can tsell them at $150, she won t sell them at all. Given that the demand for tickets isa binomial random variable with parametersn= 10 andp= 1/2, how many ticketsshould she buy in order to maximize her expected profit?Problem thatXandYare independent discrete random variableswith the same geometric PMF:pX(k) =pY(k) =p(1 p)k 1, k= 1,2,...,wherepis a scalar with 0< p <1. Show that for any integern 2, the conditionalPMFP(X=k|X+Y=n)is ,Y, andZbe independent geometric random variables with thesame PMF:pX(k) =pY(k) =pZ(k) =p(1 p)k 1,wherepis a scalar with 0< p <1. FindP(X=k|X+Y+Z=n).Hint:Try thinkingin terms of coin shops for probability books forKhours, whereKis a randomvariable that is equally likely to be 1, 2, 3, or 4. The number of booksNthat he buysis random and depends on how long he shops according to the conditional PMFpN|K(n|k) =1k,forn= 1,...,k.(a) Find the joint PMF ofKandN.(b) Find the marginal PMF ofN.(c) Find the conditional PMF ofKgiven thatN= (d) Find the conditional mean and variance ofK, given that he bought at least 2but no more than 3 books.(e) The cost of each book is a random variable with mean $30. What is the expectedvalue of his total expenditure?Hint:Condition on the events{N= 1},...,{N=4}, and use the total expectation IndependenceProblem his workplace, the first thing Oscar does every morning is to goto the supply room and pick up one, two, or three pens with equal probability 1/3. Ifhe picks up three pens, he does not return to the supply room again that day. If hepicks up one or two pens, he will make one additional trip to the supply room, wherehe again will pick up one, two, or three pens with equal probability 1/3. (The numberof pens taken in one trip will not affect the number of pens taken in any other trip.)Calculate the following:(a) The probability that Oscar gets a total of three pens on any particular day.(b) The conditional probability that he visited the supply room twice on a given day,given that it is a day in which he got a total of three pens.(c)E[N] andE[N|C], whereE[N] is the unconditional expectation ofN, the totalnumber of pens Oscar gets on any given day, andE[N|C] is the conditionalexpectation ofNgiven the eventC={N >3}.(d) N|C, the conditional standard deviation of the total number of pens Oscar getson a particular day, whereNandCare as in part (c).(e) The probability that he gets more than three pens on each of the next 16 days.(f) The conditional standard deviation of the total number of pens he gets in thenext 16 days given that he gets more than three pens on each of those computer has been acting very strangely lately, and you suspectthat it might have a virus on it. Unfortunately, all 12 of the different virus detectionprograms you own are outdated. You know that if your computer does have a virus,each of the programs, independently of the others, has a chance of believing thatyour computer as infected, and a chance of thinking your computer is fine. On theother hand, if your computer does not have a virus, each program has a chanceof believing that your computer is fine, and a chance of wrongly thinking yourcomputer is infected. Given that your computer has a chance of being infectedwith some virus, and given that you will believe your virus protection programs onlyif 9 or more of them agree, find the probability that your detection programs will leadyou to the right Lucky plays the lottery on any given week with probabilityp,independently of whether he played on any other week. Each time he plays, he has aprobabilityqof winning, again independently of everything else. During a fixed timeperiod ofnweeks, letXbe the number of weeks that he played the lottery andYbethe number of weeks that he won.(a) What is the probability that he played the lottery on any particular week, giventhat he did not win on that week?4(b) Find the conditional PMFpY|X(y|x).(c) Find the joint PMFpX,Y(x,y).(d) Find the marginal PMFpY(y).Hint:One possibility is to start with the answerto part (c), but the algebra can be messy. But if you think intuitively about theprocedure that generatesY, you may be able to guess the answer.(e) Find the conditional PMFpX|Y(x|y). Do this algebraically using the precedinganswers.(f) Rederive the answer to part (e) by thinking as follows: for each one of then Yweeks that he did not win, the answer to part (a) should tell you