10SimHw2Sol - IEOR 4404 Assignment #2 Solutions Simulation...

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Unformatted text preview: IEOR 4404 Assignment #2 Solutions Simulation February 8, 2010 Prof. Mariana Olvera-Cravioto Page 1 of ?? Assignment #2 Solutions 1. var ( c 1 X 1 + c 2 X 2 ) = E [( c 1 X 1 + c 2 X 2 ) 2 ]- E [ c 1 X 1 + c 2 X 2 ] 2 = E [ c 2 1 X 2 1 + c 2 2 X 2 2 + 2 c 1 c 2 X 1 X 2 ]- ( c 1 E [ X 1 ] + c 2 E [ X 2 ]) 2 = c 2 1 E [ X 2 1 ] + c 2 2 E [ X 2 2 ] + 2 c 1 c 2 E [ X 1 X 2 ]- c 2 1 E [ X 1 ] 2- c 2 2 E [ X 2 ] 2- 2 c 1 c 2 E [ X 1 ] E [ X 2 ] = c 2 1 ( E [ X 2 1 ]- E [ X 1 ] 2 ) + c 2 2 ( E [ X 2 2 ]- E [ X 2 ] 2 + 2 c 1 c 2 ( E [ X 1 X 2 ]- E [ X 1 ] E [ X 2 ]) = c 2 1 var ( X 1 ) + c 2 2 var ( X 2 ) + 2 c 1 c 2 cov ( X 1 X 2 ) 2. Let S be your total commute time, N be the number of red traffic lights you encounter on your way to work, and X i be the delay on the i th red light. Then, S = 15 + N summationdisplay i =1 X i . Note that N is a Binomial r.v. with parameters (7 , . 2), and E [ X 1 ] = (0 . 5 + 2) / 2 = 1 . 25, var( X 1 ) = (2- . 5) 2 / 12 = 0 . 1875. Using the conditional expectation and variance formulas: E [ S ] = 15 + E [ N ] E [ X 1 ] = (1 . 4)(1 . 25) = 16...
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10SimHw2Sol - IEOR 4404 Assignment #2 Solutions Simulation...

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