MA519_JAN11 - Math 519 B Davis Qualifying Examination...

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Math 519 – B. Davis Qualifying Examination January, 2011 Do not do arithmetic or simplify answers. (15) 1. Let X and Y be independent random variables each uniform on [0 , 1]. Give a function H ( x,y )=( u ( x,y ) ,v ( x,y )) such that H ( X,Y ) has a uniform distribution on the parallelogram with vertices (0 , 0) , (0 , 2) , (3 , 3), and (3 , 5). (20) 2. A pond contains M golden fish and K silver fish. The fish are removed one at a time at random until all the fish remaining in the pond are the same color. Find the expectation and the variance of the number of fish remaining in the pond. (15) 3. Let X i ,1 i 10, be independent exponential random variables. Le Y 1 and Y 2 be independent random variables which are both discrete uniform on { 0 , 1 , 2 } and which are also independent of the exponentials. Find the probability that of these twelve random variables, the Y j ’s are the second and third order statistics, that is the second and third smallest of the twelve numbers. (25) 4. Sunny, partly cloudy, and cloudy days each have probability 1/3 of occurring
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