6.041 Fall 2005 Quiz 1Wednesday, October 19, 7:30-9:30 PM.DO NOT TURN THIS PAGE OVER UNTILYOU ARE TOLD TO DO SOName:Recitation Instructor:TA:Question Score Out of0 31 122 143 284 43Your Grade 100• You have 12 0 minutes to complete the quiz.• Write your solutions in this exam booklet. We will not consider any work not in theexam booklet.• This quiz has five problems that are not necessarily in order of difficulty.• You may give an answer in the form of an arithmetic expression (sums, products, ratios,factorials) of numbersthat could be evaluated using a calculator. Expressions like83orP5k=0(1/2)kare also fine.• A correct answer does not guarantee full credit and a wrong answer does not guaranteeloss of credit. You should concisely indicate your reasoning and show all releva nt work.The grade on each problem is based on our judgment of yo ur level of understanding asreflected by what you have written.• This is a closed-book exam except for one double-sided, handwritten, 8.5 by 11 formulasheet.• Calculators are not allowed.• Be neat! If we can’t read it, we can’t grade it.Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 20 05)Problem 0: (3 points)Write your name, your recitation in structor’s name, and your TA’s name on the front of thebooklet.Problem 1: (12 points)Random variable H takes on the value 1 with probability 1/3, and the value of 0 with probability2/3. Random variable W is described by the conditional probabilities as follows :pW |H(w|1) =2/3, w = 01/3, w = 10, otherwisepW |H(w|0) =1/2, w = 11/2, w = 20, otherwisepW|H(w|0)0 1 2 3w1/2pW|H(w|1)0 1 2 3w2/31/3(a) (4 points) Determine the joint PMF of H and W , i.e., pH,W(h, w).pH,W(h, w) =(b) (4 points) Determine pH| W(0|1).pH| W(0|1) =(c) (4 points) Are W and H independent? Explain why or why not.Answer: Yes / NoPage 2 of 6Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 20 05)Problem 2: (14 points)Consider three events A, B and C. You know the following information about these events:• P(A) = P(B) = P(C) =110• P(A ∩ B ∩ C) =11000• P(A ∩ Bc) =9100• P(Bc|C) =45(a) (4 points) Are events A and B independent? Explain your answer.Answer: Yes / No(b) (5 points) Are events A, B and C independent? Explain your answer.Answer: Yes / No(c) (5 points) Compute P(A|B ∩ C).P(A|B ∩ C) =Page 3 of 6Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 20 05)Problem 3: (28 points)You are playing a simple card game with another friend, using a standard 52 card deck: foursuits (Diamonds, Hearts, Clubs and Spades), 13 cards of each suit (numbers from 2 to 10, Jack,Queen, King and Ace). An honest card dealer, different from you two, picks 8 cards ou t of 52 cardsat random and divides them evenly between you two, 4 cards each.(a) (5 points) What is the probability that you get Ace, King, Queen and Jack of the Diamondsuit?P(You get Ace, King, Queen and Jack of the Diamond suit) =(b) (5 points) What is the probab ility that you get Ace, King, Queen and Jack of same suit?P(You get Ace, King, Queen and Jack of the same suit) =(c) (6 points) What is the pr obab ility that you get Ace, King, Queen and Jack?P(You get Ace, King, Queen and Jack ) =(d) (6 points) What is the probab ility that all four of your cards are of the Diamond suit?P(All four of your cards are of the Diamond suit) =(e) (6 points) You have Ace, King, Qu een and Jack of Diamonds. What is the p robability thatall of the other player’s cards are of the Diamond suit?P(Other player has all diamonds | You have Ace, King, Queen and Jack of Diamonds) =Page 4 of 6Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 20 05)Problem 4: (43 points)Oscar is a used car salesman who is always trying to maximize his earnings, but often losescustomers by trying to rush their decision. His manager decides to help him spend more time witheach customer by limiting the number of customers Oscar is allowed to interact with during theday. He presents Oscar with two different options:Strategy 1: Have one distinct customer interaction per 15 minute slot, where there is a total of32 slots per day. Th e probability of a sale for each interaction is 1/4, and all customer interactionsand sales are independent of each other. Each sale yields a commission c1= $80.Strategy 2: Have one distinct customer interaction per 30 minute slot, where there is a total of16 slots per day. Th e probability of a sale for each interaction is 1/2, and all customer interactionsand sales are independent of each other. Each sale yields a commission c2= $100.Let K be the number of cars Oscar sells on a particular day using Strategy 1. His earningson that day are Q1= K · c1. Let M be the number of cars Oscar sells on a particular day usingStrategy 2. His earnings on that day are Q2= M · c2.Question (f) can be solved independently of (d) and (e). Also, question (g) can be solvedindependently of (d), (e) and (f).(a) (5 points) Determine the PMF of K.pK(k) =(b) (5 points) Find the expectation and the variance of K. Provide numerical answers .E[K] =V ar(K) =(c) (5 points) Determine the expected earnings for each strategy and state which choice Oscarshould take to maximize his expected earnings.E[Q1] =E[Q2] =Which strategy should Oscar take? Strategy 1 / Strategy 2Page 5 of 6Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 20 05)(d) (7 points) Despite the above analysis, Oscar can’t make up his mind of which choice to take.He decides to fl ip a fair coin and follow Strategy 1 if it turns up heads, or follow Strategy 2if it turns up tails. Let L be the number of cars h e sells using this strategy. Determine thePMF and the expectation of L. Provide a numerical answer for the expectation.pL(l) =E[L] =(e) (7 points) Oscar is still using his randomized strategy from part d. Given that he sold 4 carsone day, what is the probability that the coin turned up heads? If he had instead sold 18cars, what would be the probability that the coin turned up heads?P(Head | Oscar sold 4 cars) =P(Head | Oscar sold 18 cars) =(f) (7 points)
View Full Document
Unlocking...