MA519_JAN07

# MA519_JAN07 - bers of descendants Let N 2 be the population...

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MATHEMATICS QUALIFYING EXAMINATION JANUARY 2007 MATH 519 - Prof. Sellke Each problem is worth 20 points. 1. Twelve dots are arranged in four rows, with three dots in each row. Randomly choose four of the twelve dots. Let N be the number of rows with no chosen dot. Find the mean and variance of N . 2. Let X 1 and X 2 be random variables with joint density f X ( x 1 ,x 2 )= ± 3 x 1 if 0 <x 2 <x 1 < 1 0e l s e Let Y 1 = 1 X 1 and Y 2 = 1 X 2 . Find the joint density f Y ( y 1 ,y 2 )o f Y 1 and Y 2 . 3. Suppose that a solution now contains a single living bacterium. This organism has the property that, after 24 hours, it will give rise to a random number N 1 of descendants with a Geometric(p) “number of failures” distribution: P { N 1 = k } = q k p, k =0 , 1 , 2 ,... with p (0 , 1) and q =1 - p and EN 1 = q p .(So , N 1 is the population size after 24 hours.) Furthermore, each bacterium present in 24 hours will give rise after

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Unformatted text preview: bers of descendants. Let N 2 be the population size 48 hours from now. Find E ( N 1 | N 2 = 0). (Hint: The maximum value of this quantity as p varies between 0 and 1 is 1 3 .) 4. Let X and Y be independent standard normal (i.e., N (0 , 1)) random variables. Find P { 3 X 2 < Y 2 } . 5. Let U 1 ,U 2 ,...,U n be iid U [0 , 1] random variables, with order statistics ≤ U (1) ≤ U (2) ≤ ··· U ( n ) ≤ 1 . For k = 1 , 2 ,...,n + 1, let G k = U ( k )-U ( k-1) be the length of the k th “gap” (where we set U (0) = 0 and U ( n +1) = 1). Let L n = max { G k , 1 ≤ k ≤ n + 1 } be the length of the largest gap. When n = 10 43 , the median of the random variable L n is approximately an integer power of 1 10 , so that median( L 10 43 ) ≈ 10-j for some integer j . Find j , and justify your answer. (Heuristic reasoning is ﬁne.)...
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MA519_JAN07 - bers of descendants Let N 2 be the population...

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