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131A_1_FinalPractice2010Upload

131A_1_FinalPractice2010Upload - EE 131A Probability...

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EE 131A Fall 2010 Practice Final Probability December 6, 2010 Instructor: Lara Dolecek Maximum score is 200 points. You have 180 minutes to complete the exam. Please show your work. Good luck! Problem Score Possible 1 15 2 10 3 20 4 15 5 10 6 15 7 15 8 20 9 20 10 25 11 10 12 25 Total 200 1
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1. (15 pts) The random variable X has cdf as shown in Figure. Compute (a) P [ X < - 1], (b) P [ X ≤ - 1], and (c) P [ | X - 0 . 5 | > 0 . 5]. 2
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2. (10 pts) Suppose ( X, Y, Z ) are distributed as follows: ( X, Y, Z ) = (0 , 0 , 0) with probability 0 . 25 , (0 , 5 , 5) with probability 0 . 25 , (5 , 5 , 0) with probability 0 . 25 , (5 , 0 , 5) with probability 0 . 25 . (a) Are X and Y independent ? (b) Are X , Y and Z independent ? 3
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3. (20 pts) Suppose X 1 , X 2 , ... , X n are independent and uniformly distributed on [ a, b ]. Let Y = max { X 1 , X 2 , . . . , X n } . (a) Find pdf of Y . (b) What are E [ Y ] and V AR ( Y )? 4
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4. (15 pts) Fifty-two percent of the residents of New York are in favor of outlawing cigarette smoking in publicly owned places. A pollster randomly picks n people and determines what percentage of them are in favor of banning smoking. How large should n
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