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reviewmidterm2_sol

# reviewmidterm2_sol - IEOR 172 Probability and Risk Analysis...

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IEOR 172: Probability and Risk Analysis for Engineers, Fall 2007 Review Questions for Midterm2 1. The lifetime of each light bulbs is exponetially distributed with mean 5 hours. We test 12 light bulbs (one at a time) whether its quality exceeds our standard. If it lasts longer than 4 hours, we say it passes the test; otherwise, it fails. Let X be the number of light bulbs that pass the test. (a) What is the distribution of X ? Let L i denote the lifetime of light bulb i . We have L i exp(1 / 5). The probability that each light bulb passes the test is P( L i > 4) = e - 4 / 5 . The distribution of X is Binomial(12, e - 4 / 5 ). (b) Let Y denote the number of light bulbs that fail the test. What are E[ Y ] and Var( X )? We have Y = 12 - X . So E[ Y ] = E[12 - X ] = 12 - E[ X ] = 12 - 12 · e - 4 / 5 . Similarly, Var( Y ) = Var(12 - X ) = ( - 1) 2 Var( X ) = 12 · e - 4 / 5 · (1 - e - 4 / 5 ) . (c) What is the probability that X > Y ? P( X > Y ) = P( X > 12 - X ) = P( X > 6) = P( X 7) = 12 X i =7 12 i ! e - 4 / 5 i 1 - e - 4 / 5 12 - i . 2. X and Y have the following joint probability mass functions: p XY ( x, y ) = ( c e - 5 5 x x !

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