prelim2#1 - ORIE 360/560 Engineering Probability and...

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Unformatted text preview: ORIE 360/560 { Engineering Probability and Statistics II Fall 2003 Midterm 2 All problems have equal weight. SHOW YOUR WORK. GOOD LUCK! (4x + 2y + 1) if 0 < x < 1 and 0 < y < 2 : 0 otherwise (a) Compute Cov(X; Y ) and the correlation . (b) Find the number a that minimizes the variance of the linear combination aX + Y . Problem 2 A vaccine for desensitizing patients to bee stings is to be packed with three vials in each box. Each vial is checked for strength before packing. The probability that a vial meets speci cations is :9. (a) Let X denote the number of vials that must be checked to ll a box. Find the pmf of X , its mean and its variance. (b) Suppose that only 10 vials with the vaccine have been prepared. What is the probability that this is enough to ll 3 boxes? Problem 3 Describe the inverse transform method to generate a sample of continuous random variables with the triangular density 8 if 0 < x < 1=2 < 4x 4(1 x) if 1=2 x < 1 : f = :0 otherwise f X;Y is given by Problem 1 The joint density of a continuous random vector (X; Y ) (x; y) = 1 10 X;Y X Problem 4 A system consists of three independent components, A, B and C, connected in a series. The lifespan of component A has exponential distribution with parameter :2 (days) 1, the lifespan of component B has exponential distribution with parameter :1 (days) 1 and the lifespan of component C has exponential distribution with parameter :15 (days) 1. The system fails when one of the components fails. (a) Find the probability that the component whose failure caused the system to fail is component C. (b) Assume that you can increase the expected lifespan of one of the components by 1 day. Which component should you select for this improvement in order to cause the maximal possible increase in the expected lifespan of the entire system? 1 ...
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This test prep was uploaded on 04/13/2008 for the course ORIE 360 taught by Professor Ehrlichman during the Fall '07 term at Cornell.

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