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# 09_27_11 - 1 Let X 1 X 2 X n be a random sample from the...

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1. Let X 1 , X 2 , … , X n be a random sample from the distribution with probability density function ( ) 4 3 θ 4 θ x e x 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 3 1 4 θ 4 θ . By Factorization Theorem, Y = = n i i 1 4 X is a sufficient statistic for θ . OR f ( x ; θ ) = { } 4 ln ln ln exp 3 4 θ θ x x + + + - . K ( x ) = x 4 . Y = ( ) = n i i 1 X K = = n i i 1 4 X is a sufficient statistic for θ . b) What is the probability distribution of Y = = n i i 1 4 X ? F X ( x ) = 4 θ 1 x e - - , x > 0 θ > 0. F W ( w ) = P ( X 4 w ) = w e θ 1 - - , w > 0 θ > 0. W = X 4 has Exponential distribution with mean = “usual θ ” = θ 1 . Y = = n i i 1 4 X = = n i i 1 W has Gamma distribution with α = n and “usual θ ” = θ 1 .

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c) Use part (b) to suggest a confidence interval for θ with ( 1 - α ) 100 % confidence level. 2 W / “usual θ = 2 θ = n i i 1 4 X has a chi-square distribution with r = 2 α = 2 n d.f. P ( ( ) 2 2 1 2 χ
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