# 409Hw03ans - STAT 409 Fall 2009 Homework #3 (due Friday,...

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STAT 409 Homework #3 Fall 2009 (due Friday, September 18, by 4:00 p.m.) 1. Let X 1 , X 2 , … , X n be a random sample of size n from the distribution with probability density function ( 29 ( 29 ( 29 θ 2 X X ln 1 θ θ ; x x x f x f - = = , x > 1, θ > 1. a) Find the sufficient statistic Y = u ( X 1 , X 2 , … , X n ) for θ . f ( x 1 ; θ ) f ( x 2 ; θ ) f ( x n ; θ ) = ( 29 = - n i i i x x 1 θ 2 ln 1 θ = ( 29 = - = - n i i n i i n x x 1 θ 1 2 ln 1 θ . Y 1 = = n i i 1 X is a sufficient statistic for θ . Y 2 = ln Y 1 = ln = n i i 1 X = = n i i 1 X ln is also a sufficient statistic for θ . OR ( 29 ( 29 θ 2 X ln 1 θ θ ; x x x f - = = exp { θ ln x + ln ln x + 2 ln ( θ – 1 ) } K ( x ) = ln x . Y 2 = = n i i 1 X ln is a sufficient statistic for θ . Y 1 = e Y 2 = = n i i 1 X is also a sufficient statistic for θ .

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b) What is the probability distribution of W = ln X ? Using the change-of-variable technique ( Section 3.5, pp. 173 – 174 ): X continuous r.v. with p.d.f. f X ( x ). Y = u ( X ) u ( x ) one-to-one, differentiable ( strictly increasing or strictly decreasing ) X = u 1 ( Y ) = v ( Y ) f Y ( y ) = f X ( v ( y ) ) | v ' ( y ) | W = ln X X = e W = v ( W ) v ' ( w ) = e w f W ( w ) = ( θ – 1 ) 2 w e w θ e w = ( θ – 1 ) 2 w e w ( θ – 1 ) = ( 29 ( 29 2 2 1 θ Γ - w 2 – 1 e w ( θ – 1 ) , w > 0. W has Gamma ( α = 2, “usual θ ” = 1 1 θ - ) distribution. c) What is the probability distribution of = n i i 1 X ln ? Suppose X and Y are independent, X is Gamma ( α 1 , θ ), Y is Gamma ( α 2 , θ ). If random variables X and Y are independent, then M X + Y ( t ) = M X ( t ) M Y ( t ). M X + Y ( t ) = ( 29 ( 29 2 1 α α θ θ 1 1 1 1 t t - - = ( 29 2 1 α α θ 1 1 + - t , t < θ 1 . X + Y is Gamma ( α 1 + α 2 , θ ); = n i i 1 X ln = = n i i 1 W has Gamma ( α = 2 n , “usual θ ” = 1 1 θ - ) distribution.
2. Let X 1 , X 2 , … , X n be a random sample from the distribution with probability density function ( 29 ( 29 X X θ 2 θ θ ; x e x x f x f - = = x > 0 θ > 0. a) Find the sufficient statistic Y = u ( X 1 , X 2 , … , X n ) for θ . f

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## This note was uploaded on 10/12/2011 for the course STATISTICS stat 410 taught by Professor Stepanov during the Spring '11 term at University of Illinois, Urbana Champaign.

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409Hw03ans - STAT 409 Fall 2009 Homework #3 (due Friday,...

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