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Unformatted text preview: STAT 410 Homework #10 Spring 2008 (due Friday, April 4, by 3:00 p.m.) 1. Let > 0 be an unknown parameter and let X 1 , X 2 , , X n be independent random variables, each with the probability density function f ( x ) = ( ) & & < < otherwise 1 1 1 & & x x . a) Obtain the maximum likelihood estimator of , n & . b) Find the CDF of X 1 . c) Let W i = ln ( 1 X i ), i = 1, 2, , n . Find the CDF and the PDF of W 1 . d) Find E ( W 1 ) and Var ( W 1 ). e) Use WLLN ( Theorem 4.2.1 ) and Theorem 4.2.4 to show that n & is a consistent estimator for ( as n ). f) Use CLT ( Theorem 4.4.1 ) and Theorem 4.3.9 to show that n & is asymptotically normally distributed ( as n ). Find the parameters. 2. Let > 0 be an unknown parameter and let X 1 , X 2 , , X n be independent random variables, each with the probability density function f ( x ) = ( ) & & < < otherwise 1 1 1 & & x x ....
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This note was uploaded on 02/11/2011 for the course STAT 410 taught by Professor Monrad during the Spring '08 term at University of Illinois, Urbana Champaign.
 Spring '08
 Monrad
 Probability

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