class8 q

class8 q - 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 EXAMPLE CLASS 8 1. 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 . Suppose the test with critical region C k is used to test H 0 : θ 1 against H 1 : θ > 1 . (a) Find the power function of the test and show that it is an increasing function in θ . (b) Deduce from (a) the size of the test. 2. Let X 1 ,...,X n be i.i.d. Poisson random variables with mean θ . (a) Find the most powerful size α test of H 0 : θ = 1 vs H 1 : θ = 1 . 21 (b) Use the Central Limit Theorem to show that the smallest value of n required to make α = 0 . 05 and β 0 . 1 ( where α and β are the type I and II error probabilities respectively) is somewhere near 212. 3. A random variable
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class8 q - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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