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Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2008)Problem Set 41Due: October 1, 2008Topics: Variance, joint pmf’s, conditioning, conditional expectation, multiplediscrete random variables, independence. [Text sections: 2.4–2.7]1. Chuck will go shopping for probability books for K hours. Here, K is a random variable andis equally likely to be 1, 2, 3, or 4. The number of books N that he buys is random anddepends on how long he shops. We are told thatpN|K(n | k) =1k, for n = 1, . . . , k.(a) Find the joint PMF of K and N.(b) Find the marginal PMF of N.(c) Find the conditional PMF of K given that N = 2.(d) We are now told that he bought at least 2 but no more than 3 books. Find the conditionalmean and variance of K, given this p iece of information.2. Professor May B. Right often has her science facts wrong, and ans wers each of her students’questions incorrectly with probability 1/4, independently of other questions. In each lectureProfessor Right is asked either 1 or 2 questions with equal probability.(a) What is the probability that Professor Right gives wrong answers to all the questionsshe gets in a given lecture?(b) Given that Professor Right gave wrong answers to all the questions she was asked in agiven lecture, what is the probability that she got two questions?(c) Let X and Y be th e number of questions asked and the number of questions answeredcorrectly in a lecture, respectively. What are the mean and variance of X and the meanand the variance of Y ?(d) Give a neatly labeled s ketch of the joint PMF pX,Y(x, y).(e) Let Z = X + 2Y . What is the expectation of Z? Explain how you would compute thevariance of Z. You don’t need to give the numerical value of Var(Z). (Use materialfrom 9/29 lecture.)(f) Determined to improve her reputation, Professor Right decides to teach an additionalcourse in her specialty (math), where she answers questions incorrectly with probability1/10 rather than 1/4. The two courses have the same number of lectur es, and the errorsshe might make in any lecture are independent of the errors in all other lectures. Whatis the expected number of questions that she will answer wrong in a r an domly chosenlecture (math or science)?3. A pair of fair four-sided dice is thrown once. Each die has faces labeled 1, 2, 3, an d 4.Discrete rand om variable X is defined to be the product of th e down-face values. Determinethe conditional variance of X2given that the sum of the down-face values is greater than theproduct of the down-face values.1Published September 23, 2008Page 1 of 2Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2008)4. Your probability class has 250 undergraduate students and 50 graduate students. Each stu-dent is assigned a grade randomly, independently of others, with undergraduate studentsgetting an A with probability 1/3 and graduate students getting an A w ith p robability 1/2.Let X be the numb er of students that get an A.(a) Find the PMF of X.(b) Calculate E[X] using the total expectation theorem, rather than the PMF of X.(c) Let W = X + 2. Calculate E[W ].5. (Use material from 9/29 lecture.)At Tony’s pizza, the following four toppings are available: (1) mushroom, (2) sausage, (3)pepperoni and (4) onion. A random pizza has topping i with probability pi= 2−iindependentof whether that pizza has any other topping and each pizza is ordered independently of everyother pizza. On a day in which the number of pizzas sold is n, let Niequal th e number ofpizzas sold with topping i. What is the joint PMF pN1,N2,N3,N4(n1, n2, n3, n4)?G1†. (Use material from 9/29 lecture.)Let X1, . . . , Xnbe independent and identically distributed random variables. FindE[X1| X1+ · · · + Xn= x0] ,where x0is a constant.G2†. Suppose we have m jelly beans and n jars, and we throw the jelly beans randomly andindependently into the jars (i.e. each jelly bean is equally likely to end up in any of th e jars,independently of all the other jelly beans; assume each jar is large enough to hold m jellybeans).(a) Find the expected number of jelly beans in the ith jar, for 1 ≤ i ≤ n.(b) Find the expected number of empty jars.(c) Find the probability that every jar receives at least one jelly bean (assume m ≥ n).†Required for 6.431; optional for 6.041 Page 2 of


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MIT 6 041 - Problem Set 4

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