409Quiz1Bans - x ) = x 2 . Y = ( ) = n i i 1 X K = = n i i...

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STAT 409 Fall 2011 Version B 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 ( ) 2 γ 12 9 5 X γ γ ; x e x x f - = , x > 0, γ > 0. a) (3) Find a sufficient statistic Y = u ( X 1 , X 2 , … , X n ) for γ . f ( x 1 , x 2 , x n ; γ ) = f X ( x 1 ; γ ) f X ( x 2 ; γ ) f X ( x n ; γ ) = = = - n i i x n n x n i i e 1 9 5 γ 12 1 2 γ . By Factorization Theorem, Y = = n i i 1 2 X is a sufficient statistic for γ . OR f X ( x ; γ ) = { } 2 n ln ln exp l 9 12 5 γ γ x x + - + - . K (
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Unformatted text preview: x ) = x 2 . Y = ( ) = n i i 1 X K = = n i i 1 2 X is a sufficient statistic for . b) (7) Obtain the maximum likelihood estimator of , . L ( ) = =- n i x i i e x 1 9 5 2 12 = =- = n i i x n i i n n e x 1 9 5 1 2 12 ln L ( ) = = = -+-n i i n i i x x n n 1 2 1 9 12 5 ln ln ln ( ln L ( ) ) ' = =-n i i x n 1 2 5 = 0 = = n i i n 1 2 X 5 ....
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This note was uploaded on 09/14/2011 for the course STAT 409 taught by Professor Stephanov during the Fall '11 term at University of Illinois at Urbana–Champaign.

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409Quiz1Bans - x ) = x 2 . Y = ( ) = n i i 1 X K = = n i i...

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