# 10_10 - STAT 410 Examples for 10/10/2011 Fall 2011 4. Let X...

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STAT 410 Examples for 10/10/2011 Fall 2011 4. Let X 1 , X 2 , … , X n be a random sample of size n from the distribution with probability density function ( ) ( ) θ 2 X ln 1 θ θ ; x x x f - = , x > 1, θ > 1. c) What is the probability distribution of W = ln X ? Using the change-of-variable technique ( Theorem 1.7.1 ): 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 ) | Let W = ln X X = e W dw dx = e w f W ( w ) = ( θ – 1 ) 2 w e w θ e w = ( θ – 1 ) 2 w e w ( θ – 1 ) = ( ) ( ) 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. ( “usual λ ” = θ – 1 ) d) What is the probability distribution of Y = = n i i 1 X ln ? Suppose

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## This note was uploaded on 10/31/2011 for the course MATH 464 taught by Professor Monrad during the Fall '08 term at University of Illinois, Urbana Champaign.

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10_10 - STAT 410 Examples for 10/10/2011 Fall 2011 4. Let X...

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