07_22 - STAT 410 Examples for Spring 2009 Def Let X 1 X 2...

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Unformatted text preview: STAT 410 Examples for 07/22/2009 Spring 2009 Def Let X 1 , X 2 , … , X n denote random variables with joint p.d.f. or p.m.f. f ( x 1 , x 2 , … , x n ; θ ) , which depends on the parameter θ . The statistic Y = u ( X 1 , X 2 , … , X n ) is said to be sufficient for θ if the conditional distribution of X 1 , X 2 , … , X n given Y = y is independent of θ for all y . Theorem 1 ( Factorization Theorem ) : Let X 1 , X 2 , … , X n denote random variables with joint p.d.f. or p.m.f. f ( x 1 , x 2 , … , x n ; θ ) , which depends on the parameter θ . The statistic Y = u ( X 1 , X 2 , … , X n ) is sufficient for θ if and only if f ( x 1 , x 2 , … , x n ; θ ) = φ [ u ( x 1 , x 2 , … , x n ) ; θ ] ⋅ h ( x 1 , x 2 , … , x n ) , where φ depends on x 1 , x 2 , … , x n only through u ( x 1 , x 2 , … , x n ) and h ( x 1 , x 2 , … , x n ) does not depend on θ . 1. Let X 1 , X 2 , … , X n be a random sample of size n from a Poisson distribution with mean λ . That is,...
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This note was uploaded on 10/29/2009 for the course STAT 410 taught by Professor Alexeistepanov during the Summer '08 term at University of Illinois at Urbana–Champaign.

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07_22 - STAT 410 Examples for Spring 2009 Def Let X 1 X 2...

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