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Unformatted text preview: STAT 410 Homework #9 Spring 2008 (due Monday, March 31, by 4:30 p.m.) No credit will be given without supporting work. Warm-up: 4.2.3 By Chebyshevs Inequality, P ( | W n | ) 2 2 W & n = 2 & p n b 0 as n for all > 0. Therefore, W P n . 1. 4.3.2 F Y 1 ( x ) = ( ) --- x n e 1 , x > . ( Recall Homework 8, problems 6 & 7. ) F Z n ( z ) = P ( n ( Y 1 ) z ) = P ( Y 1 n z + ) = 1 e z , z > 0. Therefore, the limiting distribution of Z n is Exponential with mean 1. ( Exponential distribution with mean 1 is same as Gamma distribution with = 1, = 1. ) 2. 4.3.3 F Y n ( x ) = ( ) ( ) n x F . Since Z n = n ( 1 F ( Y n ) ), P ( Z n ) = 1. Let z > 0. F Z n ( z ) = P ( n ( 1 F ( Y n ) ) z ) = P ( F ( Y n ) 1 n z ) = & & & --- n z n 1 F F 1 1 Y = n n z 1 1 & -- 1 e z as n . For the limiting distribution Z of Z n , F Z ( z ) = 1 e z , z > 0, f Z ( z ) = e z , z > 0. Therefore, the limiting distribution of Z n is Exponential with mean 1....
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This homework help was uploaded on 04/13/2008 for the course STAT 410 taught by Professor Alexeistepanov during the Spring '08 term at University of Illinois at Urbana–Champaign.
- Spring '08