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Unformatted text preview: Solutions for suggested exercises from Ch5 page 1 of 18 From Textbook Ch5: 1, 2, 3, 4, 5, 6, 7, 8-v8, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 33, 34, 40, 42, 44, 48, 56, 71, 80, 91 5-1. The time T required to repair a machine is an exponentially distributed random variable with mean 1/2 (hours). (a) What is the probability that a repair time exceeds 1/2 hour? (b) What is the probability that a repair takes at least 12 1 2 hours given that its duration exceeds 12 hours? 5-2. Suppose that you arrive at a single-teller bank to find five other customers in the bank, one being served and the other four waiting in line. You join the end of the line. If the service times are all exponential with rate , what is the expected amount of time you will spend in the bank? 5-3. Let X be an exponential random variable. Without any computations, tell which one of the following is correct. (a) E [ X 2 | X &gt; 1] = E [( X + 1) 2 ]. (b) E [ X 2 | X &gt; 1] = E [ X 2 ] + 1. (c) E [ X 2 | X &gt; 1] = (1 + E [ X ]) 2 . 5-4: Consider a post office with two clerks. Three people, A, B and C enter simultaneously. A and B go directly to the clerks, and C waits until either A or B leaves before he begins service. What is the probability that A is still in the post office after the other two have left when (a) the service time for each clerk is exactly (non-random) ten minutes? (b) the service times are i with probability 1/3, i = 1 , 2 , 3? (c) the service times are exponential with mean 1 / ? 5-5. The lifetime of a radio is exponentially distributed with a mean of ten years. If Jones buys a ten-year-old radio, what is the probability that it will be working after an additional ten years? 5-6: Consider a post office that is run by two clerks. Suppose that when Mr. Smith enters the system he discovers that Mr. Jones is being served by one of the clerks and Mr. Brown by the other. Suppose also that Mr. Smith is told that his service will begin as soon as either Jones or Brown leaves. If the amount of time that clerk i , i = 1 , 2, spends with a customer is exponentially distributed with mean 1 . i , show that P (Smith is not last) = 1 1 + 2 2 + 2 1 + 2 2 5-7: If X 1 and X 2 are independent nonnegative continuous random variables, show that P ( X 1 &lt; X 2 | min( X 1 ,X 2 ) = t ) = r 1 ( t ) r 1 ( t ) + r 2 ( t ) , where R i ( t ) is the failure rate function of X i . 5-8. Let X i , i = 1 , 2 ,...,n be independent continuous random variables, with X i having failure rate function R i ( t ). Let T be independent of this sequence, and suppose that n i =1 P ( T = i ) = 1. Show that the failure rate function R ( t ) of X T is given by r ( t ) = n X i =1 r i ( t ) P ( T = i | X T &gt; t ) . Solutions for suggested exercises from Ch5 page 2 of 18 5-8-v8: If X has failure rate function r ( t ), show that E [ X ] = E [ 1 r ( X ) ]....
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- Spring '08