Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2010)Problem Set 3Due September 29, 20101. The hats of n persons are thrown into a box. The persons then pick up th eir hats at random (i.e.,so that every assignment of the hats to the persons is equally likely). What is the probabilitythat(a) every person gets his or her hat back?(b) the first m persons who picked hats get their own hats back?(c) everyone among the firs t m persons to pick up the hats gets back a hat belonging to one ofthe last m persons to pick up the hats?Now assum e, in addition, that every hat thrown into the box has probability p of getting dirty(independently of wh at happens to the other hats or who has dropped or picked it up). What isthe probability that(d) the first m persons will pick up clean hats?(e) exactly m persons will pick up clean hats?2. Alice plays with Bob the following game. First Alice r andomly chooses 4 cards out of a 52-carddeck, memorizes them, and places them back into the deck. Then Bob randomly chooses 8 cardsout of the same deck. Alice wins if Bob’s cards include all cards selected by her. What is theprobability of this happening?3. (a) Let X be a random variable that takes nonnegative integer values. Show thatE[X] =∞Xk=1P(X ≥ k).Hint: Express the right-hand side of th e above formula as a double s ummation then inter-change the order of the summations.(b) Use the formula in th e previous part to find the expectation of a random variable Y whosePMF is defined as follows:pY(y) =1b − a + 1, y = a, a + 1, . . . , bwhere a and b are nonnegative integers with b > a. Note that for y = a, a + 1, . . . , b, pY(y)does not depend explicitly on y since it is a uniform PMF.4. Two fair three-sided dice are rolled s imultaneously. Let X be the difference of th e two rolls.(a) Calculate the PMF, the expected value, and the variance of X.(b) Calculate and plot the PMF of X2.5. Let n ≥ 2 be an integer. Show thatnXk=2k(k − 1) nk!= n(n − 1)2n−2.Hint: As one way of solving the problem, following from Example 1.31 in the text, think of acommittee that includes a chair and a vice-chair.Page 1 of 2Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2010)G1†. A candy factory has an endless sup ply of red, orange, yellow, green, blue, black, white, and violetjelly beans. The factory packages the jelly beans into jars in such a way that each jar has 200beans, equal number of red and orange beans, equal number of yellow and green beans, on e moreblack bean than the number blue beans, and three more violet beans than the number of whitebeans. One possible color distribution, for example, is a jar of 50 yellow, 50 green, one black,48 white, and 51 violet jelly beans. As a marketing gimmick, the factory guarantees that notwo jars have the s ame color distribution. What is the maximum number of jars the factory canproduce?†Required for 6.431; optional for 6.041 Page 2 of
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