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Problem 31. lConsider four independent rolls of a Bsided die. Let X be the number of la and let Y be the number of is: obtained. \"mat is the joint PMF' of X and Y? Problem 32. D. Bernoulli's problem of joint lives. Consider 2111 persons forming in couples who live together at a given time. Suppose that at some later time, the probability of each person being alive is 1:, independent of other persons. At that later time. let A be the number of persons that are alive and let S be the number of couples in which both partners are alive. For any survivor number a, nd E[S| A = a]. Problem 31* A coin that has probability of heads equal to p is tossed successively and independently until a bead comes twice in a row or a tail comes twice in a row. Find the expected value of the number of kisses. Problem 24. A stock market trader buys 1o] shares of stock A and EDD shares of stock E Let X and Y be the price changes of A and B. respectively. over a certain time period. and assume that the joint PMF of X and Y is uniform over the set of integers :1: and y satisfying {a} Find the marginal PMFs and the means of X and Y. {b} Find the mean of the trader's profit. Problem 25. A class of 11 students takes a test consisting of m qumtions. Suppose that student I? submitted answers to the rst m. questions (a) The grader randomly picks one answer. call it (LJ), where f is the student ID number {taking values 1,....u] and J is the question number [taking values 1. . . . , m]. Assume that all answers are equally likely to be picked. Calculate the joint and the marginal PMFs of f and J. [b] Assume that an answer to question j. if submitted by student 1', is correct with probability 1t?1| j. Each answer gets a points if it is correct and gets in points otherwise. Calculate the expected value of the score of student 1'. Problem 216. PMF of the minimum of several random variables. (in a given day. your golf score takes values from the range ll to 11D. with probabity [1.1, independent of other days. Determined to improve your score. you decide to play on three different days and declare as your score the minimum X of the scores X1, X2, and Hg on the diffo'ent days. {a} Calculate the PMF of X. {b} By how much has your expected score improved as a result of playing on three days? Problem 2'3"." The multinomial distribution. A die with r faces, numbered 1, . . . . r. is rolled a xed number of times a. The probability that the ith face comes up on any one roll is denoted p\" and the results of different rolls are amumed independent. Let X, be the number of times that the ith face comes up. (a) Find the joint PMF pxi ..... xrfkl, . .. ,kr}. {b} Find the expected value and variance of I}. {c} Find E[A'.XJ] fort :p': j. Problem 20. As an advertising campaign, a chocolate factory places golden tickets in some of its candy bars, with the promise that a golden ticket is worth a trip through the chocolate factory, and all the chocolate you can eat for life. If the probability of finding a golden ticket is p, nd the mean and the variance of the number of candy bars you need to eat to nd a ticket. Problem 21. St. Petersburg paradox. You toss independently a fair coin and you count the number of tosses until the rst tail appears If this number is 71, you receive 2\" dollars. 1What is the expected amount that you will receive? How much would you be willing to pay to play this game? Problem 22. Two coins are simultaneously tossed until one of them mmes up a head and the other a tail. The rst coin comes up a head with probability p and the second with probability 9. All tosses are assumed independent. {a} Find the PMF, the expected value, and the variance of the number of tosses. {b} 1What is the probability that the last toss of the rst coin is a head? Problem 23. {a} A fair coin is tossed repeatedly and independently until two consecutive heads or two consecutive tails appear. Find the FMF, the expected value, and the variance of the number of tosses. 124 Discrete Random Variables Chap. 2 {b} Assume now that the coin is tossed until we obtain a tail that is immediately preceded by a head. Find the PMF and the expected value of the number of tosses. Problem 16. Let X be a random variable with PMF $2,311. if 1: = s,2,1,e,1,2,s, 0, otherwise. Pr: {1'} = { {a} Find a and ELK]. (b) 1What is the PMF of the random variable 3 = (X ELY\": '3' {c} Using the result from part (b), find the variance of K. {d} Find the variance oi'X using the formula vaer] = 21(1' E[X])2px[:s}. Problem 1?. A city's temperature is modeled as a random variable with mean and standard deviation both equal to 1D degrees Celsius A day is described as \"normal\" if the temperature during that day ranges within one standard deviation from the mean. What would be the temperature range for a normal day if temperature were expressed in degrees Fahrenheit? Problem 15. Let a and b be positive integers with o '5 l1, and let X be a random variable that takes as values, with equal probability, the powers of 2 in the interval [2", 2"] . Find the expected value and the variance of X. Problems 123 Problem 19. A prize is randomly placed in one of ten boxes, numbered from 1 to 1D. You search for the prize by asking yesno questions Find the expected number of questions until you are sure about the location of the prize, under each of the following strategies. {a} An enumeration strategy: you ask questions of the form \"is it in box It?" . {b} A bisection strategy: you eliminate as close in half of the remaining boxes as possible by asking questions of the form "is it in a box numbered less than or equal to H". n I 1" 'IIF +'I'I I" 'I 4" I 'I 'i l" 'I u u 'I 4* II I 'I Problem 29.* The inclusion-exclusion formula. Let A1, A2, ..., An be events. Let S1 = {ill