midterm1old

midterm1old - i-μ 1 1 2 1 n P n j =1 X 2 j Y 2 j 1 n P n k...

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ST 530 – February 28, 2005 Name: Midterm #1 All problem parts have equal weight. In budgeting your time expect that some problems will take longer than others. 1. Let X 1 , . . . , X n be an i.i.d. sample from Exponential(1) distribution. (a) Find ES 2 n , Var S 2 n . (Hint: When calculating μ 4 think Γ functions). (b) Find the density of X b n/ 2 c : n . (c) What is the asymptotic (approximate) distribution of ¯ X n and X b n/ 2 c : n ? (d) Does X b 3 n/ 4 c : n - X b n/ 4 c : n P -→ a ? If yes, ﬁnd the value of a . (e) Is there a sequence a n such that X n : n - a n D -→ Y ? If yes, ﬁnd it and ﬁnd the distribution of Y . (Hint: Finding the c.d.f. of Y is suﬃcient.)

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2. Assume that ± X Y ² , ± X 1 Y 1 ² , ± X 2 Y 2 ² , . . . are i.i.d. random vectors, where both X and Y are positive random variables with ﬁnite moments of all orders. Denote the raw moments μ 0 r,s = EX r Y s , r = 0 , 1 , 2 . . . , s = 0 , 1 , 2 , . . . . (a) Find the limiting distribution of n ( 1 n n i =1 X i Y i - μ 0 1 , 1 ) . (b) Find the limiting distribution of n ( 1 n P n i =1 X i Y
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Unformatted text preview: i-μ 1 , 1 ) 2 ( 1 n P n j =1 X 2 j Y 2 j )-( 1 n P n k =1 X k Y k ) 2 . (Hint: Use Slutsky’s theorem.) (c) Find the approximate distribution of ( ¯ X n )( ¯ Y n ). (Hint: Use the multivariate delta method.) 3. (a) Let X and Y be independent continuous random variables each of them having a density belonging to an exponential family. Prove or disprove: The random vector ( X, Y ) has a joint density belong-ing to an exponential family. (b) Let ( X, Y ) be jointly continuous random variables having a joint density belonging to an exponential family. Prove or disprove: The marginal distribution of X has a density belonging to an exponential family. (Hint: Consider a joint density in the form g ( θ ) e θxy I (0 , 1) ( x ) I (0 , 1) ( y ). )...
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midterm1old - i-μ 1 1 2 1 n P n j =1 X 2 j Y 2 j 1 n P n k...

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