{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

This preview shows pages 1–4. Sign up to view the full content.

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 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 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 ( x 1 , x 2 , x n ; θ ) = f ( x 1 ; θ ) f ( x 2 ; θ ) f ( x n ; θ ) = = = - n i i x n n x n i i e 1 1 1 2 θ θ .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}