Tutorial 7

Tutorial 7 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1302 PROBABILITY AND STATISTICS II (2009-10) EXAMPLE CLASS 7 1. Let X have a pdf of the form f ( x ) = θx θ - 1 , 0 < x < 1 , where θ ∈ { θ : θ = 1 , 2 } . To test the simple hypothesis H 0 : θ = 1 against the alternative simple hypothesis H 1 : θ = 2, use a random sample X 1 , X 2 of size n=2 and deﬁne the critical region to be C = { ( x 1 ,x 2 ) : 3 / 4 x 1 x 2 } . Find the power function of the test. 2. Let X 1 and X 2 be i.i.d. exponential random variables with mean θ . It is well known that S = X 1 + X 2 has a gamma distribution with the density function f ( s ) = s exp( - s/θ ) 2 , s > 0 . Deﬁne a region C k = { ( x 1 ,x 2 ) : x 1 + x 2 > k } for some positive k . (a) Suppose that the test with critical region C k is used to test H 0 : θ = 1 against H 1 : θ = 2 . i. Show that the Type I error probability of the test is (1 + k )exp( - k ). ii. Find the Type II error probability of the test.

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This note was uploaded on 05/04/2011 for the course STAT 1302 taught by Professor Smslee during the Spring '10 term at HKU.

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Tutorial 7 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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