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6.041/6.431 Fall 2009 Quiz 1Tuesday, October 13, 12:05 - 12:55 PM.DO NOT TURN THIS PAGE OVER UNTILYOU ARE TOLD TO DO SOName:Recitation Instructor:TA:Question Score Out ofA 2B.1 10B.2 (a) 10B.2 (b i) 12B.2 (b ii) 12B.2 (c) 10B.3 (a) 10B.3 (b) 12B.3 (c) 12B.3 (d i) 5B.3 (d ii) 5Your Grade 100• This quiz has 2 problems, worth a total of 100 points.• You may tear apart pages 3 and 4, as per your convenience.• Write your solutions in this quiz booklet, only solutions in this quiz booklet will be graded.Be neat! You will not get credit if we can’t read it.• You are allowed one two-sided, handwritten, 8.5 by 11 formula sheet. Calculators are notallowed.• Parts B.2 and B.3 can be done independently.• You may give an answer in the form of an arithmetic expression (sums, products, ratios,factorials) of numbers that could be evaluated using a calculator. Expressions like83orP5k=0(1/2)kare also fine.• You have 50 minutes to complete the quiz.• Graded quizzes will be returned in recitation on Thursday 10/15.Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2009)Problem A: (2 points) Write your name, your assigned recitation instructor’s name, and assignedTA’s name on the cover of the quiz booklet. The Instructor/TA pairing is listed below.Recitation Instructor TA Recitation TimeJeffrey Shapiro Jimmy Li 10 & 11 AMDanielle Hinton Uzoma Orji 1 & 2 PMWilliam Richoux Ulric Ferner 2 & 3 PMJohn Wyatt (6.431) Aliaa Atwi 11 & 12 PMPage 2 of 11Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2009)116 Discrete Random Variables Chap. 2of a discrete random variable. We showed how to use the PMF of a randomvariable X to calculate the mean and the variance of a related random variableY = g(X) without calculating the PMF of Y . In the special case where g isa linear function, Y = aX + b, the means and the variances of X and Y arerelated byE[Y ]=aE[X]+b, var(Y )=a2var(X).We also discussed several special random variables, and derived their PMF,mean, and variance, as summarized in the table that follows.Summary of Results for Special Random VariablesDiscrete Uniform over [a, b]:pX(k)=!1b − a +1, if k = a, a +1,...,b,0, otherwise,E[X]=a + b2, var(X)=(b − a)(b − a + 2)12.Bernoulli with Parameter p: (Describes the success or failure in a singletrial.)pX(k)="p, if k = 1,1 − p, if k = 0,E[X]=p, var(X)=p(1 − p).Binomial with Parameters p and n: (Describes the number of successesin n independent Bernoulli trials.)pX(k)=#nk$pk(1 − p)n−k,k=0, 1, . . . , n,E[X]=np, var(X)=np(1 − p).Geometric with Parameter p: (Describes the number of trials until thefirst success, in a sequence of independent Bernoulli trials.)pX(k) = (1 − p)k−1p, k =1, 2,...,E[X]=1p, var(X)=1 − pp2.Page 3 of 11Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2009)Problem B: (98 points) As a way to practice his probability skills, Bob goes apple picking. Theorchard he goes to grows two varieties of apples: gala and honey crisp.The proportion of the gala apples in the orchard is p (0 < p < 1), the proportion of thehoney crisp apples is 1 − p. The number of apples in the orchard is so large that you can assumethat picking a few apples does not change the proportion of the two varieties.Independent of all other apples, the probability that a randomly picked gala apple is ripe is gand the probability that a randomly picked honey crisp apple is ripe is h.1. (10 points) Suppose that Bob picks an apple at random (uniformly) and eats it. Find theprobability that it was a ripe gala apple.Note: Parts 2 and 3 below can be done independently.2. Suppose that Bob picks n apples at random (independently and uniformly).(a) (10 points) Find the probability that exactly k of those are gala apples.(b) Suppose that there are exactly k gala apples among the n apples Bob picked. Calebcomes by and gives Bob a ripe gala apple to add to his bounty. Bob then picks an appleat random from the n + 1 apples and eats it.(i) (12 points) What is the probability that it was a ripe apple?(ii) (12 points) What is the probability that it was a gala apple if it was ripe?(c) (10 points) Let n = 20, and suppose that Bob picked exactly 10 gala apples. What is theprobability that the first 10 apples that Bob picked were all gala?3. Next, Bob tries a different strategy. He starts with a tree of the gala variety and picks apples atrandom from that tree. Once Bob picks an apple off the tree, he carefully examines it to makesure it is ripe. Once he comes across an apple that is not ripe, he moves to another gala tree.He does this until he encounters an unripe apple on that second tree. Assume that each tree hasa very large, essentially infinite, number of apples.(a) (10 points) Let Xibe the number of apples Bob picks off the ith tree, (i = 1, 2). Writedown the PMF, expectation, and variance of Xi.(b) (12 points) For i = 1, 2, let Yibe the total number of ripe apples Bob picked from thefirst i trees. Find the expectation and the variance of Y2. (Note that Y1= X1− 1 andY2= (X1− 1) + (X2− 1).)(c) (12 points) Find the joint PMF of Y1and Y2.(d) In the following, answer just “yes” or “no.” (Explanations will not be taken into accountin grading.)(i) (5 points) Are X1and Y2independent?(ii) (5 points) Are X2and Y1independent?Each question is repeated in the following pages. Please write your answer onthe appropriate page.Page 4 of 11Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2009)1. (10 points) Suppose that Bob picks an apple at random (uniformly) and eats it. Find theprobability that it was a ripe gala apple.Note: Parts 2 and 3 can be done independently.2. Suppose that Bob picks n apples at random (independently and uniformly).(a) (10 points) Find the probability that exactly k of those are gala apples.Page 5 of 11Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2009)(b) Suppose that there are exactly k gala apples among the n apples Bob picked. Calebcomes by and gives Bob a ripe gala apple to add to his bounty. Bob then picks an appleat random from the n + 1 apples and eats it.(i)


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MIT 6 041 - EXAM- 6.041

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