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409Quiz1Cans - x = x 3 ⇒ Y = ∑ = n i i 1 X K = ∑ = n...

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STAT 409 Fall 2011 Version C Name ANSWERS . Quiz 1 (10 points) Be sure to show all your work, your partial credit might depend on it. No credit will be given without supporting work. 1. Let X 1 , X 2 , … , X n be a random sample from the distribution with probability density function ( ) 3 α 2 11 4 X α α ; x e x x f - = , x > 0, α > 0. a) (3) Find a sufficient statistic Y = u ( X 1 , X 2 , … , X n ) for α . f ( x 1 , x 2 , x n ; α ) = f X ( x 1 ; α ) f X ( x 2 ; α ) f X ( x n ; α ) = = = - n i i x n n x n i i e 1 11 4 α 2 1 3 α . By Factorization Theorem, Y = = n i i 1 3 X is a sufficient statistic for α . OR f X ( x ; α ) = { } 3 n ln ln exp l 11 2 4 α α x x + - + - . K ( x )
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Unformatted text preview: x ) = x 3 . ⇒ Y = ( ) ∑ = n i i 1 X K = ∑ = n i i 1 3 X is a sufficient statistic for α . b) (7) Obtain the maximum likelihood estimator of α , α ˆ . L ( α ) = ∏ =- n i x i i e x 1 11 4 3 α 2 α = ∑ =-∏ = n i i x n i i n n e x 1 11 4 1 3 α 2 α ln L ( α ) = ∑ ∑ = = ⋅ ⋅ ⋅-+-n i i n i i x x n n 1 3 1 11 2 4 α α ln ln ln ( ln L ( α ) ) ' = ∑ =-n i i x n 1 3 4 α = 0 ⇒ α ˆ = ∑ = n i i n 1 3 X 4 ....
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