� � 6.041/6.431 Spring 2006 Quiz 1 Wednesday, March 15, 7:30-9:30 PM. DO NOT TURN THIS PAGE OVER UNTIL YOU ARE TOLD TO DO SO Name: Recitation Instructor: TA: Question Score Out of 0 3 1 30 2 40 3 27 Your Grade 100 • You have 120 minutes to complete the quiz. • At the end of the quiz period you will turn in this quiz packet, and 2 blue books. Question 1 will be answered in the quiz packet, while question 2 and 3 will be answered in their own respective blue books. • Question 1 is True False, no partial credit is given. • For questions 2 and 3, you should concisely indicate your reasoning and show all relevant work. Grades will be based on our judgment of your level of understanding as reflected by what you have written. • You may give an answer in the form of an arithmetic expression (sums, products, ratios, 8factorials) of numbers that could be evaluated using a calculator. Expressions like 3 �5 or k=0(1/2)k are also fine. • This is a closed-b ook exam except for one double-sided, handwritten, 8.5 by 11 formula sheet. • Calculators are allowed. • Be neat! If we can’t read it, we can’t grade it.Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2006) Problem 0: (3 points) Write your name, your recitation instructor’s name, and TA’s name on the cover of the quiz booklet and your 2 blue books. Include the question number on the cover of the blue book. Problem 1: (30 points) Each of the following statements is either True or False. There will be no partial credit given for the True False questions, thus any explanations will not be graded. Please clearly indicate True or False in the below, ambiguous marks will receive zero credit. All questions have equal weight. Consider a probabilistic model with a sample space Ω, a collection of events that are subsets of Ω, and a probability law P() defined on the collection of events—all exactly as usual. Let A, B and C be events. (a) If P(A) ≤ P(B), then A ⊆ B. True False (b) Assuming P(B) > 0, P(A | B) is at least as large as P(A). True False Now let X and Y be random variables defined on the same probability space Ω as above. (c) If E[X] > E[Y ], then E[X2] ≥ E[Y2]. True False (d) Suppose P(A) > 0. Then E[X]= E[X | A]+ E[X | AC]. True False (e) If X and Y are independent and P(C) > 0, then pX,Y |C(x, y)= pX|C(x) pY |C(y). True False (f) If for some constant c we have P({X>c})= 1 2, then E[X] >c 2. True False In a simple game involving flips of a fair coin, you win a dollar every time you get a head. Suppose that the maximum number of flips is 10, however, the game terminates as soon as you get a tail. (g) The expected gain from this game is 1. True False Let X be a uniformly distributed continuous random variable over some interval [a, b]. (h) We can uniquely describe fX(x) from knowing its mean and variance. True False Let X be an exponentially distributed random variable with a probability density function fX(x)= −xe. (i) Then P ({0 ≤ X ≤ 3}∪{2 ≤ X ≤ 4})=1 − e−4 True False Let X be a normal random variable with mean 1 and variance 4. Let Y be a normal random variable with mean 1 and variance 1. (j) P(X<0) < P(Y<0). True FalseMassachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2006) Problem 2: (40 points) Please write all work for Problem 2 in your first blue book. No work recorded below will be graded. All questions have approximately the same weight. Borders Book store has been in business for 10 years, and over that period, the store has collected transaction data on all of its customers. Various marketing teams have been busy using the data to classify customers in hopes of better understanding customer spending habits. Marketing Team A has determined that out of their customers, 1/4 are low frequency buyers (i.e., they don’t come to the store very often). They have also found that out of the low frequency buyers, 1/3 are high spenders (i.e., they spend a significant amount of money in the store), whereas out of the high frequency buyers only 1/10 are high spenders. Assume each customer is either a low or high frequency buyer, and each customer is either a high or low spender. (a) Compute the probability that a randomly chosen customer is a high spender. (b) Compute the probability that a randomly chosen customer is a high frequency buyer given that he/she is a low spender. You are told that the only products Borders sells are books, CDs, and DVDs. You are in-troduced to Marketing Team B which has identified 3 customer groupings. These groups are collectively exhaustive and mutually exclusive. They have also determined that each customer is equally likely to be in any group, customers are i.i.d (independent and identically distributed), and each customer buys only one item per day. They refer to the groupings as C1, C2,and C3,and have determined the following conditional probabilities: P(purchases a book | customer in C1) = 1/2 P(purchases a CD | customer in C1) = 1/4 P(purchases a DVD | customer in C1) = 1/4 P(purchases a book | customer in C2) = 1/2 P(purchases a CD | customer in C2)=0 P(purchases a DVD | customer in C2) = 1/2 P(purchases a book | customer in C3) = 1/3 P(purchases a CD | customer in C3) = 1/3 P(purchases a DVD | customer in C3) = 1/3 (c) Compute the probability that a customer purchases a book or a CD. (d) Compute the probability that a customer is in group C2 or C3 given he/she purchased a book. Now in addition to the data from Marketing Team B, you are told that each book costs $15, each CD costs $10, and each DVD costs $15. (e) Compute the PMF, expected value and variance of the revenue (in dollars) Borders collects from a single item purchase of one customer?Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2006) (f) Suppose that n customers shop on a given day. Compute the
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