tut04

STAT 2011 Probability and Estimation Theory – Semester 1, 2023 Tutorial Sheet Week 4 Tutorial Problems for Week 4 1. Suppose a series of n independent trials can end in one of three possible outcomes. Let k 1 and k 2 denote the number of trials that result in outcomes 1 and 2, respectively. Let p 1 and p 2 denote the probabilities associated with outcomes 1 and 2. Deduce a formula for the probability of getting k 1 and k 2 occurrences of outcomes 1 and 2, respectively. 2. Repair calls for central air conditioners fall into three general categories: coolant leakage, compressor failure, and electrical malfunction. Experience has shown that the probabili- ties associated with the three are 0.5, 0.3, and 0.2, respectively. Suppose that a dispatcher has logged in ten service requests for tomorrow morning. Use the answer to Question 3.2.18 to calculate the probability that three of those ten will involve coolant leakage and five will be compressor failures. 3. Find the probability mass function (pmf) for the discrete random variable X whose cumu- lative distribution function (cdf) at the points x = 0 , 1 , . . . , 6 is given by F X ( x ) = x 2 / 36. 4. Consider the following game. A fair coin is flipped until the first head appears. The win is 2 if it appears on the first toss, 4 if it appears on the second toss, and, in general, 2 k if it first occurs on the k th toss. Let the random variable X denote the winnings. How much must the bet be in order for this to be a fair game? [Note: A fair game is one where the difference E [ X n +1 − X n ] = 0.] 5. An urn contains four chips numbered 1 through 4. Two are drawn without replacement. Let the random variable X denote the larger of the two. Find E ( X ). 6. Suppose the experiment consists of tossing 3 fair coins. Let Y denote the number of heads that appear, define the pmf p Y ( k ). 7. Independent trials consisting of the flipping of a coin having probability p of coming up heads are continually performed until either a head occur or a total of n flips is made. Let X denote the number of times the coin is flipped, find p X ( k ), and check that it is a pmf. 8. Find E( X ) and Var( X ), where X is the outcome when we roll a fair die. 9. A school class of 120 students is driven in 3 buses to a symphonic performance. There are 36 students in one of the buses, 40 in another, and 44 in the third bus. When the buses arrive, one of the 120 students is randomly chosen. Let X denote the number of students on the bus of that randomly chosen student, and find E( X ). 10. Suppose there are m days in a year, and that each person is independently born on day r with probability p r , with r = 1 , . . . , m and ∑ m k =1 p k = 1. Let A i,j be the event that person i and person j are born on the same day. (a) Find P ( A 1 , 3 ). (b) Find P ( A 1 , 3 | A 1 , 2 ). (c) Show that P ( A 1 , 3 | A 1 , 2 ) ≥ P ( A 1 , 3 ). 1