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STAT 333 tutorial2

# STAT 333 tutorial2 - total number of eggs in a ﬁeld of...

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Stat 333 - Tutorial 2 1. Boxes of cereal contain a promotional toy. The toys are equally likely to be any of k action figures. If a person purchases n boxes of cereal, let X denote the number of distinct action figures that have not been acquired. Find E ( X ) and V ar ( X ) . 2. (Chapter 5 Problem #9 of the textbook.) Machine 1 is currently working. Machine 2 will be put in use at a time t from now. If the lifetime of machine i is exponential with rate λ i , i = 1 , 2 , what is the probability that machine 1 is the first machine to fail? 3. X and Y are independent random variables, each with p.d.f. p (1 - p ) n - 1 , n = 1 , 2 , . . . . Find an expression for P ( X Y ) . 4. (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
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Unformatted text preview: total number of eggs in a ﬁeld of the speciﬁed area. 5. (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 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. 1...
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