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Assign 2 - 3/08 THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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Unformatted text preview: 3/08 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1302 PROBABILITY AND STATISTICS II Assignment 2 (Sketch Solution) 1. In what follows, X stands for the random data, x stands for the observed data, θ stands for the parameter, Θ stands for the parameter space and ‘ ( θ ) stands for the likelihood function. (a) (i) x = 100. (ii) θ = N , Θ = { 2000 , 2001 ,... } . (iii) ‘ ( θ ) = ( 1000 100 )( N- 1000 1900 ) / ( N 2000 ) . (b) (i) x = 60. (ii) θ = p , Θ = [0 , 1]. (iii) ‘ ( θ ) = ( 100 60 ) p 60 (1- p ) 40 . (c) (i) x = (8 , 4 , 3 , 8 , 6 , 1) (ii) Θ = (0 , ∞ ). (iii) ‘ ( θ ) = θ 5 exp ‡- θ ∑ 5 i =1 x i · e- θx 6 = θ 5 e- 30 θ . (d) (i) x = (12 , 6 , 36 , 18 , 12 , 24). (ii) θ = ( p s ,p m ,p n ,q s ,q m ,q n ), Θ = { ( p s ,p m ,p n ,q s ,q m ,q n ) ∈ [0 , 1] 6 : ∑ p = ∑ q = 1 } . (iii) ‘ ( θ ) = { 54! / (12!6!36!) } p 12 s p 6 m p 36 n { 54! / (18!12!24!) } q 18 s q 12 m q 24 n . (e) (i) x = 3. (ii) θ = λ , Θ = (0 , ∞ ). (iii) ‘ ( θ ) = λ 3 e- λ / 3!....
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Assign 2 - 3/08 THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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