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Assignment 2- solution

# Assignment 2- solution - THE UNIVERSITY OF HONG KONG...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1302 PROBABILITY AND STATISTICS II (2009-10) Assignment 2 solutions 1. (a) i. Observed data: number of marked fish in recaptured load x = x = 100. ii. Parameter: θ = N ∈ { 2900 , 2901 ,... } . iii. Likelihood function: ‘ ( θ ) = ( 1000 100 )( N- 1000 1900 ) ( N 2000 ) . (b) i. Observed data: number of heads turned up x = x = 60. ii. Parameter: θ = p ∈ [0 , 1]. iii. Likelihood function: ‘ ( θ ) = ( 100 60 ) p 60 (1- p ) 40 . (c) i. Observed data: interarrival times of customers x = ( x 1 ,x 2 ,x 3 ,x 4 ,x 5 ,x 6 ) = (8 , 4 , 3 , 8 , 6 , 1). ii. Parameter: θ ∈ (0 , ∞ ). iii. Likelihood function: ‘ ( θ ) = θ 5 e- θ ∑ 5 i =1 x i e- θx 6 = θ 5 e- 30 θ . (d) i. Observed data: number of patients receive ordinary and new treatments x = ( x 1 ,x 2 ,x 3 ,x 4 ,x 5 ,x 6 ) = (12 , 6 , 36 , 18 , 12 , 24). ii. Parameter: θ = ( p s ,p m ,p n ,q s ,q m ,q n ) ∈ [0 , 1] 6 , ∑ p = ∑ q = 1. iii. Likelihood function: ‘ ( 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. Observed data: number of bus breakdowns x = x = 3. ii. Parameter: θ = λ ∈ (0 , ∞ ). iii. Likelihood function: ‘ ( θ ) = λ 3 e- λ 3! = λ 3 e- λ 6 . (f) i. Observed data: heights of students x = ( x 1 ,x 2 ,x 3 ,x 4 ,x 5 ,x 6 ,x 7 ,x 8 ,x 9 ) = (1 . 72 , 1 . 76 , 1 . 80 , 1 . 65 , 1 . 56 , 1 . 48 , 1 . 62 , 1 . 55 , 1 . 58). ii. Parameter: θ = ( μ 1 ,μ 2 ,σ 1 ,σ 2 ) ∈ R 2 (0 , ∞ ) 2 . iii. Likelihood function: ‘ ( θ ) = 1 ( √ 2 πσ 1 ) 4 exp (- ∑ 4 i =1 ( x i- μ 1 ) 2 2 σ 2 1 ) 1 ( √ 2 πσ 2 ) 5 exp (- ∑ 9 i =5 ( x i- μ 2 ) 2 2 σ 2 2 ) = 1 ( √ 2 π ) 9 σ 4 1 σ 5 2 exp ‰- 12 . 0185- 13 . 86 μ 1 + 4 μ 2 1 2 σ 2 1- 12 . 1473- 15 . 58 μ 2 + 5 μ 2 2 2 σ 2 2 ....
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Assignment 2- solution - THE UNIVERSITY OF HONG KONG...

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