quiz solution09fall - Corr ( W,Z ) = Cov ( W,Z ) / p V ar (...

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ISyE 6413(Fall 2009): “ASSESS YOUR BACKGROUND” Quiz Solution Problem 1. 1. We have 26 letters and 10 digits, so the total number is 26 5 * 10 = 118813760 . 2. 26 5 * 5 = 59406880 . 3. 25 5 * 10 = 97656250 . Problem 2. The event that the sample will contain no more than 2 defective units can be decom- posed into 3 disjoint events: 1. sample contains 0 defective unit, with a probability of (4 / 5) 4 = 256 / 625 . 2. sample contains 1 defective unit, with a probability of (1 / 5) * (4 / 5) 3 = 64 / 625 . 3. sample contains 2 defective units, with a probability of (1 / 5) 2 * (4 / 5) 2 = 16 / 625 . The final answer is the sum of the three, which is 336 / 625 . Problem 3. Cov ( W,Z ) = E ( W - E ( W ))( Z - E ( Z )) = E ( X + Y - E ( X ) - E ( Y ))(2 X - E (2 X )) = 2 E ( X + Y - 6)( X - 2) = 2 E ( X 2 - 2 X + XY - 2 Y - 6 X + 12) = 2 E ( X 2 ) - 2 E ( X ) + E ( XY ) - 2 E ( Y ) - 6 E ( X ) + 12 = 229 - 2 * 2 + E ( X ) E ( Y ) - 2 * 4 - 6 * 2 + 12 = 229 - 4 + 2 * 4 - 2 * 4 - 6 * 2 + 12 = 50 V ar ( W ) = E ( W 2 ) - ( E ( W )) 2 = E ( X 2 + Y 2 + 2 XY ) - ( E ( X ) + E ( Y )) 2 = E ( X 2 ) + E ( Y 2 ) + 2 E ( XY ) - (2 + 4) 2 = 29 + 52 + 2 * 2 * 4 - 36 = 61 V ar ( Z ) = E ( Z 2 ) - ( E ( Z )) 2 = E (4 X 2 ) - (2 EX ) 2 = 4 * 29 - 4 * 4 = 100 By definition,we have
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Unformatted text preview: Corr ( W,Z ) = Cov ( W,Z ) / p V ar ( W ) V ar ( Z ) = 50 / 61 * 100 = 0 . 64 1 Problem 4. 1. P ( X > = 750 , 000) = P (( X-500 , 000) / 150 , 000 > = (750 , 000-500 , 000) / 150 , 000) = P ( Z > = 5 / 3) = 0 . 0478 where Z is a standard normal distribution variable and the probability can be found from the normal distribution table or using standard statistical software. 2. Let X be the amount of water supplied. We then have P ( X > = X ) = P (( X-500 , 000) / 150 , 000 > = ( X-500 , 000) / 150 , 000) = P ( Z > = ( X-500 , 000) / 150 , 000) = 0 . 01 Again from the standard normal distribution table or use software, we have ( X-500 , 000) / 150 , 000 = 2 . 33 . Thus X = 849500 . 2...
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This note was uploaded on 12/25/2011 for the course ISYE 6413 taught by Professor Staff during the Spring '08 term at Georgia Institute of Technology.

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quiz solution09fall - Corr ( W,Z ) = Cov ( W,Z ) / p V ar (...

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