University of California, Berkeley, Statistics 134: Concepts of ProbabilityMichael Lugo, Spring 2011Final examMay 10, 2011, 7:10 - 10:00 pmName:Student ID:This exam consists of twelve pages: this cover page; ten pages containg problems; anda table of the normal distribution. You may use a calculator, and notes on three sides of astandard 8.5-by-11-inch sheet of paper which you have written by hand, yourself. You mustshow all work other than basic arithmetic.Write your name at the top of each page.DO NOT WRITE BELOW THIS LINEQuestion: 1a 1b 1c 1d 2a 2b 2c 3a 3b 3c 3dPoints available: 5 5 5 5 6 6 6 5 5 5 5Your score:Question: 4a 4b 4c 4d 4e 4f 4g 5a 5b 5c 6a 6b 6c 6dPoints available: 4 4 4 4 4 6 4 5 10 10 5 5 5 5Your score:Question: 7a 7b 7c 8a 8b 8c 9a 9b 9cPoints available: 5 5 15 5 5 7 10 5 10Your score:1Name:1. [20] I have a well-shuffled deck of twenty cards, labelled 1, 2, . . . , 20. Five cards aredealt from this deck.For this problem, give your answer to each part as a single fraction or decimal,containing no binomial coefficients or factorials.(a) [5] Find the probability that exactly two of these five cards are labelled with a number13 or higher.(b) [5] Find the variance of the number of cards dealt which have even numbers on them.(c) [5] Find the probability that the largest of the five numbers drawn is 18 or less.(d) [5] Find the probability that the number 11 is drawn, given that the smallest numberdrawn is 3.2Name:2. [18] In a certain town, 20 percent of households have no pets, 40 percent have one pet,32 percent have two pets, and 8 percent have three pets. Assuming that pets are equallylikely to be cats or dogs:(a) [6] Give the joint distribution table of the number of cats and dogs that a randomlychosen family has.(b) [6] Find the expected number of dogs that a random family has, given that it has nocats.(c) [6] Cats are equally likely to be male or female. Dogs are also equally likely to bemale or female. What is the probability that a given household has at least two pets of thesame species (cat or dog) and opposite sexes (male and female)? (Therefore, if the ownersof these pets are not careful, they will end up with more pets.)3Name:3. [20] I have a six-sided die with its sides labeled 1, 2, 2, 3, 3, 3. I roll the die 25 times.(a) [5] Find the expected sum of all the rolls.(b) [5] Find the standard deviation of the sum of all the rolls.(c) [5] Find the approximate probability that the sum of all the rolls is greater than 60.(d) [5] Find the approximate probability that more 2s than 3s are rolled.4Name:4. [30] A point (X, Y ) is chosen uniformly at random from the triangle with vertices at(0, 0), (0, 2), and (2, 0).(a) [4] Find the marginal densities of X and Y .(b) [4] Find E(X) and E(Y ).(c) [4] Find SD(X) and SD(Y ).(d) [4] Find the conditional distribution of Y , given X = x. You can either specifya named distribution with any necessary parameters or explicitly write down the densityfunction.5Name:Problem 4, continued from previous page.(e) [4] Find E(Y |X).(f) [6] Find E(XY ).(g) [4] Find Corr(X, Y ).6Name:5. [25] Let T have the gamma(2, λ) distribution. If T = t, let U have the uniformdistribution on [0, t].(a) [5] Find E(U).(b) [10] Find V ar(U).(c) [10] Find P (U < 1/λ).7Name:6. [20] Let X and Y be independent standard normal random variables.(a) [5] Find P (X > 0|X + Y > 0).(b) [5] Find E((2X + 3Y )(X − 2Y )).(c) [5] Find Corr(aX + bY, bX + aY ), where a and b are real numbers.(d) [5] Find P (X/Y > 1).8Name:7. [25] Let X be a random variable with exponential(λ) distribution.(a) [5] Give a formula for E(Xk) which is valid for all positive integers k.(b) [5] Give a formula for V ar(Xk) which is valid for all positive integers k.(c) [15] Let R be the radius of a random circle, with exponential(1) distribution. Let Abe the area of the same random circle, so A = πR2. Find Corr(R, A).9Name:8. [17] Let X, Y, Z be independent and identically distributed random variables withmean 1 and variance 2.(a) [5] Find V ar(2X + 3Y + 4Z).(b) [5] Find V ar(XY Z).(c) [7] One of V ar(X(Y + Z)) and V ar(X(Y + X)) can be computed from the giveninformation. The other can not be. Compute the one which can be computed, and explainwhy the other cannot be computed from the information given.10Name:9. [25] X is a random variable defined by the following process. First I flip a coin; itcomes up heads with probability 1/2. If the coin comes up heads, then I set X = X1, whereX1has the Poisson distribution with parameter 1. If the coin comes up tails, then I setX = X3, where X3has the Poisson distribution with parameter 3.(a) [10] I carry out this process and I tell you X = 2. You are asked to guess if the coincame up heads or tails. What do you guess? What is the probability that your guess iscorrect?(b) [5] Find V ar(X).(c) [10] This process is carried out many times, and each time you guess whether the coincame up heads or tails. What proportion of the time will you be right? You may assumethat if X ≤ 1 you guess heads and if X ≥ 3 you guess