midterm1oldkey

# midterm1oldkey - ST 530 Name Midterm#1 All problem parts...

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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.) Solution: (a) ES 2 n = 1, Var S 2 n = 1 n ( 9 - n - 3 n - 1 ) . (b) Set k = b n/ 2 c , f X ( k ) ( x ) = n ! ( k - 1)!( n - k )! (1 - e - x ) k - 1 e - ( n - k +1) x I (0 , ) ( x ) . (c) ¯ X n as. N (1 , 1 n ). Since ξ 1 / 2 = log 2 we have X b n/ 2 c : n as. N

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midterm1oldkey - ST 530 Name Midterm#1 All problem parts...

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