hw09_sol

# hw09_sol - Homework 9 December 7 2007 Problem 1 Let X i be...

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Unformatted text preview: Homework 9 December 7, 2007 Problem 1. Let X i be the number of pulls. X i follows a geometric distribution with parameter p. Thus the likelihood function is f ( p ) = (1- p ) ∑ X i- 20 p 20 ( i = 1 , 2 , ··· , 20 ) log f ( p ) = ( ∑ X i- 20)log(1- p ) + 20log p ∂ log f ( p ) ∂p = ( ∑ X i- 20)(- 1) 1- p + 20 p Set ∂ log f ( p ) ∂p = 0 ⇒ ˆ p = 20 ∑ X i In this case, ˆ p = 20 354 = 0 . 05649 Problem 2. L ( λ ) = Y f ( x i ; λ ) = [ λ α T ( α ) ] n Y x α- 1 i e- λ ∑ x i log L ( λ ) = nα log λ- n log T ( α ) + ( α- 1) X log x i- λ X x i solve d log L ( λ ) dλ = nα λ- X x i = 0 get λ ML = nα ∑ x i Problem 3. Since Y = AX + b , then Y 1 = a 11 X 1 + a 12 X 2 + b 1 and Y 2 = a 21 X 1 + a 22 X 2 + b 2 So E Y 1 = b 1 , E Y 2 = b 2 , and Var Y 1 = a 2 11 + a 2 12 , Var Y 2 = a 2 21 + a 2 22 Cov ( Y 1 ,Y 2 ) = Cov ( Y 2 ,Y 1 ) = Cov ( a 11 X 1 + a 12 X 2 ,a 21 X 1 + a 22 X 2 ) = a 11 a 21 Cov ( X 1 ,X 1 ) + a 11 a 22 Cov ( X 1 ,X 2 ) + a 12 a 22 Cov (...
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hw09_sol - Homework 9 December 7 2007 Problem 1 Let X i be...

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