Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2008)Problem Set 3Due: September 24, 2008Topics: Counting, Discrete random variables, probability mass functions,expectations. [Text sections: 1.6; 2.1–2.4]Note: Problem set solutions are due at the start of the lecture on the date noted above, and must be turned in within15 minutes of the start of the lecture, i.e. by 12:15 pm, at the latest. Late solutions will not be accepted beyondthis time. There will be no exceptions to this rule (other than in the event of an emergency, in which case a Dean’s letter willbe required). Therefore, it is better to hand in whatever you have done by that time!1. A parking lot consists of a single row containing n parking spaces (n ≥ 2). Mary arrives whenall spaces are free. Tom is the next person to arrive. Each person makes an equally likelychoice among all available spaces at th e time of arrival. Calculate the p robability that theparking spaces selected by Mary and Tom have at most 1 empty space between them.2. A deck of 52 cards is distributed between 4 players (as in bridge).(a) Find the probab ility that player 1 gets all 13 spades.(b) Find the probability that some player gets all 13 spades.(c) Consider the following two events:i. player 1 gets all 13 spades;ii. player 1 gets the king of hearts.Are these events independent? Are they mutually exclusive?(d) Consider the following 2 events:i. all of player 1’s cards are of the same suit;ii. player 1 gets the king of hearts.Are these two events independent?3. Consider an election in which votes are cast sequentially. Candidate A receives n votes andcandidate B receives m votes, where n > m. Assuming that all of the(n+m)!n! m!orderings ofthe votes are equally likely, let Pn,mdenote the probability that A is always ahead in thecounting of the votes.(a) Compute P2,1, P3,1, P3,2, P4,1, P4,2, P4,3.(b) Find Pn,1, Pn,2.(c) Derive a recursion for Pn,min terms of Pn−1,mand Pn,m−1by conditioning on whoreceives the last vote.4. Two fair three-sided d ice are rolled simultaneously. The three faces of each die are labeled“1,” “2,” and “3,” respectively. Let X be the sum of the two r olls.(a) Calculate the PMF, the expected value, and the variance of X.(b) Calculate and plot the PMF of X2.Published September 16, 2008 Page 1 of 2Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2008)5. Fischer and Spassky play a chess match whereby the first player to win a game wins thematch. After 10 successive draws, the match is declared drawn. Each game is won by Fischerwith probability 0.4, by Spassky with probability 0.3, and is a draw with probability 0.3.(a) What is the pr obability that Fischer win s the match?(b) Calculate and plot the PMF of the duration of the match.(c) What is the expected value of the duration of the match?6. Suppose you wish to estimate a random variable X by some constant ˆx. There are manyways to measure how good of an estimate ˆx is. Here you will derive an important propertyof minimum mean-squared error estimation.Define the mean-squared estimation error bye (ˆx) = Eh(X − ˆx)2i.(This is a deterministic function of the real variable ˆx.) Show that e (ˆx) is minimized byˆx = E[X].G1†. A candy factory has an endless supply of red, orange, yellow, green, blue, and violet jellybeans. Th e factory packages the jelly beans into jars of 100 jelly beans each. One possiblecolor distribution, for example, is a jar of 56 red, 22 yellow, and 22 green jelly beans. As amarketing gimmick, the factory guarantees that no two jars have the same color distribution.(a) What is the maximum number of jars the factory can produce?(b) If a jar is chosen uniformly at random from all the jars produced by the factory (ascounted in part (a) above), what is the pr obab ility that the jar contains no yellow jellybeans?(c) If a jar is chosen uniformly as in part (b) above, what is the probability that the jardoes not have a full assortment of jelly bean colors?(d) If a jar is chosen uniformly as in part (b) above, what is the prob ability that the jarcontains jelly beans of only five of the six colors?†Required for 6.431; optional for 6.041 Page 2 of
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