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lecture2solutions

# lecture2solutions - ISYE 6414 Spring 2009 Solution 1 The...

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Unformatted text preview: ISYE 6414 - Spring 2009 Solution 1 - The Linear Model: OLS Regression Problem 1. Maximun Likelihood estimators If Y i ∼ Normal ( β + β 1 x i ,σ 2 ) , i = 1 , 2 ,...n , and Cov ( Y i ,Y j ) = 0, i 6 = j ; then the likelihood function is: L ( Y 1 ,...,Y n | β ,β 1 ) = ( 2 πσ 2 )- n 2 exp- ∑ n i =1 ( Y i- β- β 1 x i ) 2 2 σ 2 ! Writing Y = ( Y 1 ,...,Y n ), the log-likelihood can be expressed as: l ( Y | β ,β 1 ) =- n 2 ln ( 2 πσ 2 )- ∑ n i =1 ( Y i- β- β 1 x i ) 2 2 σ 2 (1) d l ( Y | β ,β 1 ) dβ = 0 ⇒ n X i =1 ( Y i- β- β 1 x i ) = 0 (2) d l ( Y | β ,β 1 ) dβ 1 = 0 ⇒ n X i =1 x i ( Y i- β- β 1 x i ) = 0 (3) Then, from equations (2) and (3) we get: n X i =1 Y i = nb + b 1 n X i =1 x i (4) n X i =1 Y i x i = b n X i =1 x i + b 1 n X i =1 x 2 i (5) And these are the normal equations for the OLS estimators. Problem 1.5 Pag.33 No, we know that the error term i is random. That implies that E ( Y i ) = β + β 1 x i + E ( i ) = β + β 1 x i and Y i = β + β 1 x i + i Problem 1.6 Pag.33 PLot... β = 200: The expected value of the response Y when X = 0 is 200. β 1 = 5: When X increases by one, the response Y is expected to increase by 5. Problem 1.7 Pag.33 a. In model (1.1) the exact probability cannot be determined unless some probability distribution is used to model the random term . b. In model (1.24), we know that i ∼ N ( ,σ 2 = 25), then: P (195 ≤ Y ≥ 205 | x = 5) = P (- 1 ≤ Z ≥ 1) Where Z ∼ N (0 , 1). Problem 1.8 Pag.33 Yes, the mean of the new observation Y * given X = 45 will still be E ( Y * ) = 9 . 5 + 2 . 1 · 45 = 104. However, the value of Y * is nor necessary 108 given that is a random vaiable....
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lecture2solutions - ISYE 6414 Spring 2009 Solution 1 The...

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