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prelim2#2

# prelim2#2 - the system to fail is component B b The system...

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ORIE 360/560 – Engineering Probability and Statistics II Fall 2005 Midterm 2 All problems have equal weight. SHOW YOUR WORK. You have 90 minutes. GOOD LUCK! Problem 1 In baseball, the World Series lasts until one team wins 4 games. This year Chicago White Sox meet Houston Astros in the World Series. A Houston fan considers the Astros to be the superior team, with a probability of . 7 of winning any single game with the Sox. ( a ) What is the probability that the Astros will win the World Series? ( b ) What is expected number of the games played in the World Series this year? Problem 2 A system consists of two independent components, A and B, connected in a series. The lifespan of component A has exponential distribution with the mean 5 days, and the lifespan of component B has exponential distribution with the mean 10 days. The system fails when one of the two components fails. ( a ) Find the probability that the component whose failure caused the system to fail is component B.
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Unformatted text preview: the system to fail is component B. ( b ) The system management needs to install a third component, also connected in a series to the other two components (so that the new system will fail when one of the three components fails). The management can install a component whose lifespan has exponential distribution with the mean x days at a cost C ( x ) = x 1 / 2 for x between . 5 and 10 days. Let T be the lifespan of the new system. Which choice of x will minimize the cost per expected lifetime C ( x ) /ET ? Problem 3 Describe the inverse transform method to generate a sample of continuous random variables with the symmetric Fr´ echet distribution F X ( x ) = b 1 2 ( 1-e 1 /x ) if x < 1 2 ( 1 + e-1 /x ) if x > . Problem 4 Let X be a Gumbel random variable, i.e. a continuous random variable with a density f X ( x ) = e-( x + e-x ) ,-∞ < x < ∞ . Let Y = X 2 . Find the pdf of Y . 1...
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