midtermsol

midtermsol - UCSD ECE153 Handout#27 Prof Young-Han Kim...

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Unformatted text preview: UCSD ECE153 Handout #27 Prof. Young-Han Kim Tuesday, May 10, 2011 Solutions to Midterm (Total: 100 points) There are 3 problems, each problem with multiple parts, each part worth 10 points. Your answer should be as clear and readable as possible. 1. Lottery (20 pts). There are two types of lottery tickets. A regular ticket has a random number distributed uniformly from [0 , 1], while a super (more expensive) ticket has a random number distributed uniformly from [0 , 2]. Suppose that Alice buys two regular ticket and Bob buys one super ticket. We will denote by X 1 and X 2 the numbers on Alice’s tickets and by Y the number on Bob’s ticket. We assume that X 1 , X 2 , and Y are independent of each other. The person with the largest number will win the lottery. (a) Find the pdf of the largest number Alice has, i.e., f U ( u ) where U = max { X 1 , X 2 } . (b) Find the probability that Alice will win the lottery, i.e., P { max { X 1 , X 2 } ≥ Y } . Solution: (a) First we compute the cdf of U : F U ( u ) = P { U ≤ u } = P { max { X 1 , X 2 } ≤ u } = P { X 1 ≤ u, X 2 ≤ u } = P { X 1 ≤ u } P { X 2 ≤ u } (since X 1 and X 2 are independent) = if u < , u 2 if0 ≤ u ≤ 1 , 1 otherwise....
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This note was uploaded on 09/16/2011 for the course ECE ECE153 taught by Professor Younghankim during the Spring '11 term at UCSD.

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midtermsol - UCSD ECE153 Handout#27 Prof Young-Han Kim...

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