Quiz2 - 1 n i i W X = = and W p n = . (a). Please derive...

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AMS312.01 Quiz 2 Spring 2010 ♠♣♥♦ Name: ________________________________ ID: ____________________ Signature: ________________________ Instruction: This is a close book exam. Anyone who cheats in the exam shall receive a grade of F. Please provide complete solutions for full credit. The exam goes from 1:00-1:40pm. Good luck! 1. Suppose a random sample of size n is drawn from a normal population with mean μ and variance σ 2 , where σ 2 is known. Please. (a). Find the maximum likelihood estimators of μ (b). Find the method of moment estimators of μ Solutions 1. (a) [i] 2 2 ( ) 1 2 2 2 2 2 2 ( ) 1 ( ) (2 ) exp , , 1, , 2 2 i x i i i x f x e x R i n μ σ πσ - - - - = = - = L [ii] likelihood function 1 2 1 ( , , , ) ( ) n n i i L f x x x f x = = = = L 1 2 2 2 2 1 2 2 1 2 2 ( ) (2 ) exp[ ] 2 ( ) (2 ) exp 2 n i i n n i i x x - = - = - = - - = - [iii] log likelihood function 2 2 1 2 ( ) ln ( )ln(2 ) ln 2 2 2 n i i x n n l L π = - = = - - - [iv] 1 2 2 ( ) 0 2 n i i x dl d = - = = ˆ X = 1. (b) ( ) E X X = = ˆ X =
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2. Let X i , i = 1, …, n, denote the outcome of a series of n independent trials, where X i = 1 with probability p, and Xi = 0 with probability (1- p). Let
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Unformatted text preview: 1 n i i W X = = and W p n = . (a). Please derive the method of moment estimator of p. (b). Please derive the maximum likelihood estimator of p. Solutions 2. (a) ( 29 ( 29 [ ] . . . 2 2 2 2 2 2 ~ ( ) , ( ) 1* 0*(1 ) ; ( ) 1 * 0 *(1 ) (1 ); i i d i i i i i X Bernoulli p E X p p p Var X E X E X p p p p p p p = + - = = - = + - - = - = - Therefore the first population mean is ( ) i E X p = And the first sample mean is: 1 n i i X W p n n = = = Set them to be equal and we found the moment estimator of p to be: W p n = . 2. (b) [i] 1 ( ) (1 ) , 1, , i i x x i f x p p i n-=-= L [ii] ( ) (1 ) i i x n x i L f x p p- = =- [iii] ln ( )ln ( )ln(1 ) i i l L x p n x p = = +-- [iv] 1 i i x n x dl dp p p-=-=- i X W p n n = = is the MLE of p...
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Quiz2 - 1 n i i W X = = and W p n = . (a). Please derive...

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