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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 ( k1) 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 ) ≈ 10j for some integer j . Find j , and justify your answer. (Heuristic reasoning is ﬁne.)...
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 Spring '09
 Math, Standard Deviation, Variance, Probability theory, probability density function, Randomness, Cumulative distribution function

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