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STAT 333 tutorial3

# STAT 333 tutorial3 - (a Find P A the probability that A is...

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Stat 333 - Tutorial 3 Amongst the questions in this 3rd tutorial are questions 4 and 5 from the 2nd tutorial. These questions are reproduced here for convenience. 1. (Adapted from J.G. Kalbfleisch. Probability and statistical inference. Number v. 1 in Springer texts in statistics. Springer-Verlag, 1985. Section 5.7 Problem #3.) The eggs of a certain insect are found in clusters. The number of eggs per cluster has a Poisson distribution with mean value μ . The probability of finding y clusters in a field of specified area is (1 - p ) y - 1 p for y = 1 , 2 , . . . , where 0 < p < 1 . Find the mean and variance of the total number of eggs in a field of the specified area. 2. (Chapter 5 Problem #20 of the textbook.) Consider a two-server system in which a cus- tomer is served first by server 1, then by server 2, and then departs. The service times at server i are exponential random variables with rates μ i , i = 1 , 2 . When you arrive, you find server 1 free and two customers at server 2 - customer A in service and customer B waiting in line. (a) Find
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Unformatted text preview: (a) Find P A , the probability that A is still in service when you move over to server 2. (b) Find P B , the probability that B is still in the system when you move over to server 2. (c) Find E [ T ] , where T is the time that you spend in the system. Hint: Write T = S 1 + S 2 + W A + W B where S i is your service time at server i , W A is the amount of time you wait in queue while A is being served, and W B is the amount of time you wait in queue while B is being served. 3. (Chapter 3 Problem #42 of the textbook.) A total of 11 people, including you, are invited to a party. The times at which people arrive at the party are independent uniform (0 , 1) random variables. (a) Find the expected number of people who arrive before you. (b) Find the variance of the number of people who arrive before you. 4. A fair die is rolled repeatedly. Let T 54 = the number of rolls until the numbers 5 and 4 ﬁrst appear consecutively in that order. Find E [ T 54 ] . 1...
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