Solution_Mid1Sp07

Solution_Mid1Sp07 - NAME: _ STAT 427 Prof. Goel Midterm I...

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NAME: ___________________________ 1 of 4 pages 4/30/2007 STAT 427 Midterm I Spring, 2007 Prof. Goel MWF 9:30 - 10.20 REMARKS: § This is a closed book examination. However, you are allowed to use your own calculator as well as two (8.5” x 11” two-sided) crib-sheets. § There are four problems worth a total of 60 points. § Please write your name on all four sheets. § You must explain each step in your answer to get partial credits. Good Luck! Problem 1. [15 points] The probability density function of the time X (in minutes) that a flight from Columbus to Detroit arrives earlier or later than its scheduled arrival is given by f ( x ) = - . 0 , 6 6 for ) x - (36 2 elsewhere x c The negative values of x indicate flight arriving early, while positive values of x indicate flight arriving late. (a) [5 points] Find the constant c in the above density. Note that f(x) = f(-x ), therefore the above pdf is symmetric around zero. Now 6 66 3 22 - 60 0 33 (36 - ) 2 - 2 (3 6 -) 3 12 2 [(36* 6 6 ) 0 ] 2 *6 * 288 Now 28 8 1 1/288. x c x d x c x d xcx c cc == = - - =⇒= ∫∫ (b) [4 points] Find the Expected value of X. [ ] ( ) . E X x fx dx = However, since the pdf is symmetric around 0, E[X] = 0. (c) [6 Points] Find the probability that one of these flights will arrive between 1 to 3 minutes earlier than its scheduled arrival.
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Solution_Mid1Sp07 - NAME: _ STAT 427 Prof. Goel Midterm I...

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