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Berkeley STAT 134 - Exam 2

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University of California, Berkeley, Statistics 134: Concepts of ProbabilityMichael Lugo, Fall 2011Exam 2November 4, 2011, 12:10 pm - 1:00 pmName: Student ID:This exam consists of seven pages: this cover page; five pages containing problems; anda table of the normal distribution. You may use a calculator, and notes on one side of astandard 8.5-by-11-inch (or A4) sheet of paper which you have written by hand, yourself.You must show all work other than basic arithmetic. (You are allowed to use a calculatorto do arithmetic, including binomial coefficients, factorials, and Φ, but you should indicatewhat you’re asking the calculator to do.) Give your answer to each part as a fraction ordecimal; your answers should not contain any binomial coefficients, factorials, or Φ.Write your name at the top of each page.The total number of possible points is 42. The number of points for each problem areindicated in brackets.DO NOT WRITE BELOW THIS LINE1. /6 4. /92. /7 5. /123. /8 TOTAL /42Figure 1. The heroes of Problem 4.1Name:1. [6 = 2+2+2] I have a coin which comes up heads with probability 1/3, and you havea coin which comes up heads with probability 1/4. We both flip our coins at the same time.We repeat this procedure infinitely many times, always flipping our two coins simultaneously.Find:(a) the probability that I get heads for the 3rd time on the 7th flip.(b) the probability that we both get our first head on the kth flip, as a function of k.(c) the probability that we both get our first head on the same flip.2Name:2. [7 = 3+4] Let X have exponential distribution with rate λ.(a) What is the PDF of Y = X2?(b) What is V ar(X2)?3Name:3. [8 = 1 + 2 + 2 + 3] Alice and Bob broke up. Alice wants to go somewhere to thinkand clear her head, so she goes fishing. There are two kinds of fish where Alice is fishing:herring and mackerel. On average, Alice gets four bites from herring per hour, and two bitesfrom mackerel per hour. Fishes of each type bite according to a Poisson process, and thePoisson processes corresponding to the two types of fish are independent.(a) What is the probability that the first two fish Alice catches are herring? Circle youranswer; no explanation is necessary.1/3 2/5 4/9 1/2 2/3(b) What is the probability that, after 2 hours, Alice has gotten exactly 10 bites total?For parts (c) and (d), let Hkdenote the time at which the kth bite from a herring occurs.Let Mkdenote the time at which the kth bite from a mackerel occurs. Let Fkdenote thetime at which the kth bite from a fish (of either type) occurs. Find:(c) E(H40− M20), exactly.(d) P (H70≤ F100), approximately.4Name:4. [9 = 2+2+2+3] Theo the monkey and Colby the hippo are friends. The zoo is boring,so they pass the time by gambling. The game they play is as follows: they roll a single fairsix-sided die. Theo pays Colby $ 1 every time a 1 is rolled, and $ 6 every time a 6 is rolled.Colby pays Theo $ 3 every time a 3 is rolled, and $ 5 every time a 5 is rolled.(a) Let Theo’s net winnings on a single roll be a random variable X. Give a table of thedistribution of X.(b) Find E(X) as a single fraction or decimal.(c) Find V ar(X) as a single fraction or decimal.(d) What is the approximate probability that Theo is ahead after 272 rolls?5Name:5. [12 = 2 + 2 + 2 + 3 + 3] Let X be a random variable with density c/x4for x ≥ 1.(a) What is c?(b) What is E(X)?(c) What is V ar(X)?(d) Let X1, X2be independent, and let them have the same distribution as X. LetZ = max(X1, X2). What is P (Z < z)?(e) What is


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Berkeley STAT 134 - Exam 2

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