oldfinal1 - UCSD ECE153 Prof Young-Han Kim Handout#33...

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UCSD ECE153 Handout #33 Prof. Young-Han Kim Tuesday, June 1, 2010 Final Examination (Fall 2008) 1. Order statistics. Let X 1 , X 2 , X 3 be independent and uniformly drawn from the interval [0 , 1]. Let Y 1 be the smallest of X 1 , X 2 , X 3 , let Y 2 be the median (second smallest) of X 1 , X 2 , X 3 , and let Y 3 be the largest of X 1 , X 2 , X 3 . For example, if X 1 = . 3 , X 2 = . 1 , X 3 = . 7, then Y 1 = . 1 , Y 2 = . 3 , Y 3 = . 7. The random variables Y 1 , Y 2 , Y 3 are called the order statistics of X 1 , X 2 , X 3 . (a) What is the probability P { X 1 X 2 X 3 } ? (b) Find the pdf of Y 1 . (c) Find the pdf of Y 3 . (d) (Difficult.) Find the pdf of Y 2 . (Hint: Y 2 y if and only if at least two among X 1 , X 2 , X 3 are y .) 2. Fair coins. We are given two coins: Coin 1 with bias (=probability of heads) 1 / 2 and Coin 2 with random bias P Unif[0 , 1]. We pick one at random and flip it three times independently. The value of the bias does not change during the sequence of tosses. Let X be the number of heads.
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