Assignment 2- solution

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

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

Page1 / 5

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online