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# HW4Sol - ISyE 2027B Probability with Applications Fall 2010...

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ISyE 2027B Probability with Applications Fall 2010 Homework 4 Solutions Q1. (a) 5 points Figure 1: The density function of X in Q1 . 0 1 2 3 . . 1/4 3/4 (b) 15 points We need to consider the different intervals. For b < 0, F ( b ) = b -∞ f ( x ) dx = 0. For 0 b 1, F ( b ) = b -∞ f ( x ) dx = b 0 3 / 4 dx = 3 4 b . For 1 b 2, F ( b ) = F (1) = 3 4 because f ( x ) = 0 between 1 and 2. For 2 b 3, F ( b ) = 1 0 f ( x ) dx + b 2 1 / 4 dx = F (1) + 1 4 ( b - 2) = b/ 4 + 1 / 4. For b > 3, F ( b ) = F (3) = 1 since f ( x ) = 0 for x > 3. Figure 2: The distribution function of X in Q1 . 0 1 2 3 . F(b) = 3/4 F(b) = 1 F(b) = 0 Q2 (10 points) For 0 x 1, f ( x ) = d dx (2 x 2 - x 4 ) = 4 x - 4 x 3 . f ( x ) = { 4 x - 4 x 3 0 x 1 0 elsewhere 1

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ISyE 2027B Probability with Applications Fall 2010 Q3 (a) (15 points) We need -∞ f ( x ) dx = 1 - 1 c (1 - x 2 ) dx = 1, i.e., c ( x - 1 3 x 3 ) | 1 - 1 = c [(1 - 1 3 - ( - 1 - 1 3 ( - 1) 3 )] = 4 3 c = 1 Thus c = 3 / 4. (b) (15 points) For b < - 1, F ( b ) = 0. For - 1 b < 1, F ( b ) = - 1 -∞ f ( x ) dx + b - 1 3 4 (1 - x 2 ) dx = - 1 4 b 3 + 3 4 b + 1 2 . For b 1, F ( b ) = 1. Figure 3: The density function of X in Q3 -3 -2 -1 0 1 2 3 0 0.2 0.4 0.6 0.8 Q4 (a) (10 points) If the passenger arrives in the following time intervals (in minutes starting
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HW4Sol - ISyE 2027B Probability with Applications Fall 2010...

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