409Hw11 - STAT 409 Homework #11 Fall 2009 (due Wednesday,...

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Homework #11 Fall 2009 (due Wednesday, November 13, by 4:00 p.m.) 1. Let λ > 0 and let X 1 , X 2 , … , X n be independent random variables, each with the probability density function f ( x ) = < + 1 0 1 1 λ λ x x x . We wish to test H 0 : λ = 1 vs. H 1 : λ > 1. a) Find a sufficient statistic for λ . b) Find a uniformly most powerful rejection region. That is, find a rejection region that is most powerful for testing H 0 : λ = 1 vs. H 1 : λ = λ 1 for all λ 1 > 1. Hint: It should look like “Reject H 0 if Y c or “Reject H 0 if Y c ”, where Y = u ( X 1 , X 2 , … , X n ) is a sufficient statistic for λ . 2. 1. (continued) Let X 1 , X 2 be a random sample of size n = 2 from a probability distribution with p.d.f. f ( x ). c) Sketch a typical rejection region obtained in part 1 (a). Hint: Recall that x 1 1, x 2 1, so c > 1 ( if you are using = n i i x 1 ). d)
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This note was uploaded on 10/12/2011 for the course STATISTICS stat 410 taught by Professor Stepanov during the Spring '11 term at University of Illinois, Urbana Champaign.

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409Hw11 - STAT 409 Homework #11 Fall 2009 (due Wednesday,...

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