HW8sol - ORIE 3500/5500 Fall Term 2008 Assignment...

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ORIE 3500/5500 Fall Term 2008 Assignment 8-Solution 1. Suppose that the printer goes down after T hours from 9 a.m. Then f T ( t ) = e - t/ 4 / 4 for t 0. (a) Let X be the indicator random variable X = ± 1 if the printer is working at 1 p.m. 0 otherwise . Now E ( X | T ) = 1 if T > 4 and for T 4, E ( X | T ) = Z 4 - T 0 e - x dx = 1 - e - (4 - T ) . Hence the required probability is E ( X ) = E [ E ( X | T )] = Z 4 0 [1 - e - (4 - t ) ] e - t/ 4 4 dt + Z 4 1 · e - t/ 4 4 dt. = Z 0 e - t/ 4 4 dt - ( e - 4 / 4) Z 4 0 e 3 t/ 4 dt = 1 - ( e - 4 / 4)(4 / 3)( e 3 - 1) = 1 - ( e - 1 - e - 4 ) / 3 = 0 . 8940 . (b) Let X be the indicator random variable X = ± 1 if the printer is working between 1 and 2 p.m. 0 otherwise . Now E ( X | T ) = 1 if T > 5, E ( X | T ) = 0 if 4 < T 5 and for T 4, E ( X | T ) = Z 4 - T 0 e - x dx = 1 - e - (4 - T ) . Hence the required probability is E ( X ) = E [ E ( X | T )] = Z 4 0 [1 - e - (4 - t ) ] e - t/ 4 4 dt + Z 5 4 0 · e - t/ 4 4 dt + Z 5 1 · e - t/ 4 4 dt.
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HW8sol - ORIE 3500/5500 Fall Term 2008 Assignment...

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