Assignment2 Solution - THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1302 PROBABILITY AND STATISTICS II(2010-11

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DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1302 PROBABILITY AND STATISTICS II (2010-11) Assignment 2 suggested 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 2 1 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 ² 1 ( 2 πσ 2 ) 5 exp ± - 9 i =5 ( x i - μ 2 ) 2 2 σ 2 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 ³ . (g) i. Observed data: number of male x = x = 1. ii. Parameter: θ = m ∈ { 1 , 2 ,..., 1996 } . 1
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This note was uploaded on 11/17/2011 for the course MUSIC 1001 taught by Professor Dr.yun during the Spring '11 term at Princess Sumaya University for Technology.

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Assignment2 Solution - THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1302 PROBABILITY AND STATISTICS II(2010-11

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