Probability2010aug - , prove that X + Y has a Poisson...

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Qualifying Exam Probability August, 2010 1. Let X i , i 1 be IID and set N = inf { n 1 : X n > X 1 } (inf φ = ). Prove that P ( N > n ) 1 /n for n 1, and use this to show EN = . 2. Let X i , i 1 be independent with P ( X i = i ) = P ( X i = - i ) = 1 2 i α and P ( X i = 0) = 1 - 1 i α . Set S n = X 1 + ...X n . (a) Prove that S n converges almost surely if and only if α > 1. (b) If α < 1, prove that n - (3 - α ) / 2 S n converges in distribution to a normal distribu- tion. Identify the mean and standard deviation of the limit. 3. Assume that X has a Poisson distribution with parameter λ , thus P ( X = k ) = e - λ λ k k ! , k = 0 , 1 ,.... (a) Find the characteristic function of X . (b) If Y is independent of X and has a Poisson distribution with parameter
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Unformatted text preview: , prove that X + Y has a Poisson distribution with parameter + . 4. Let U 1 ,...,U n be independent and uniformly distributed on (0 , 1). Let U n 1 ,...,U nn be the order statistics of U 1 ,...,U n , that is an arrangement of U 1 ,...,U n in increasing order. Thus U n 1 U nn . (a) Prove that nU n 1 converges in distribution as n and identify the limit. (b) Prove that nU n 2 converges in distribution as n and identify the limit....
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This note was uploaded on 06/19/2011 for the course MATH 680 taught by Professor Na during the Spring '11 term at University of North Carolina School of the Arts.

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