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prelim1_sol

prelim1_sol - ORIE 360/560 – Engineering Probability and...

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Unformatted text preview: ORIE 360/560 – Engineering Probability and Statistics II Solution of Fall 2006 Prelim 1 Problem 1 Let A = { the chosen die is crooked } , B = { the chosen die shows a 4 two times in a row } . We are looking for the conditional probability P ( A | B ). By the Bayes rule P ( A | B ) = P ( B | A ) P ( A ) P ( B | A ) P ( A ) + P ( B | A c ) P ( A c ) . We know that P ( A ) = 3 / 10, P ( A c ) = 7 / 10, P ( B | A ) = 1, P ( B | A c ) = (1 / 6) 2 = 1 36 . Substituting, we obtain P ( A | B ) = 3 10 · 1 3 10 · 1 + 7 10 · 1 36 = . 939 . Problem 2 ( a ) The density has to integrate to 1. Therefore, 1 = integraldisplay ∞-∞ f X ( x ) dx = integraldisplay- 1 ( x + 1) dx + integraldisplay 2 c dx = 1 2 + 2 c . Solving for c we obtain c = 1 / 4. ( b ) The general formula for the cdf is F X ( x ) = integraltext x-∞ f X ( y ) dy . For x <- 1 this results in F X ( x ) = 0. For- 1 ≤ x < 0 this gives us F X ( x ) = integraldisplay x- 1 ( y + 1) dy = 1 2 ( x + 1) 2 ....
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prelim1_sol - ORIE 360/560 – Engineering Probability and...

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