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409Quiz1Dans - ⇒ K x = x ⇒ Y = ∑ = n i i 1 X K = ∑...

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STAT 409 Fall 2011 Version D 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 ( ) x e x x f λ 240 2 6 X λ λ ; - = , x > 0, λ > 0. a) (3) Find a sufficient statistic Y = u ( X 1 , X 2 , … , X n ) for λ . f ( x 1 , x 2 , x n ; λ ) = f X ( x 1 ; λ ) f X ( x 2 ; λ ) f X ( x n ; λ ) = = = - n i i x n n x n i i e 1 2 6 240 1 λ λ . By Factorization Theorem, Y = = n i i 1 X is a sufficient statistic for λ . OR f X ( x ; λ ) = { } n ln λ ln λ exp l 2 240 6 x x + - + -
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Unformatted text preview: . ⇒ K ( x ) = x . ⇒ Y = ( ) ∑ = n i i 1 X K = ∑ = n i i 1 X is a sufficient statistic for λ . b) (7) Obtain the maximum likelihood estimator of λ , λ ˆ . L ( λ ) = ∏ =- n i x i i e x 1 2 6 λ 240 λ = ∑ =-∏ = n i i x n i i n n e x 1 2 6 1 λ 240 λ ln L ( λ ) = ∑ ∑ = = ⋅ ⋅ ⋅-+-n i i n i i x x n n 1 1 2 240 6 λ ln ln λ ln ( ln L ( λ ) ) ' = ∑ =-n i i x n 1 6 λ = 0 ⇒ λ ˆ = ∑ = n i i n 1 X 6 ....
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409Quiz1Dans - ⇒ K x = x ⇒ Y = ∑ = n i i 1 X K = ∑...

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