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Stats 100A Fall 07 Midterm

Stats 100A Fall 07 Midterm - STAT 100B Midterm Solution...

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STAT 100B Midterm Solution Problem 1 Suppose X 1 , X 2 , ..., X n Exp( λ ). The density function of Exp( λ ) is f ( x ) = λe - λx for x 0, and f ( x ) = 0 for x < 0. (1) Suppose we want to estimate λ by solving the estimating equation Pr( X > 1) = m/n , where m is the number of X i that are greater than 1. Please find the estimate of λ . (2) Please find the MLE of λ . 1. Pr ( X > 1) = R 1 λe - λx = - e - λx | 1 = e - λ = m/n ˆ λ = log( n/m ) 2. MLE L ( λ ) = λ n e - λ x i l ( λ ) = n log λ - λ x i ∂l ∂λ = n λ - x i = 0 ˆ λ = 1 / ¯ X Problem 2 (1) Suppose X 1 , X 2 , ..., X n N( μ, σ 2 ). Please find the MLE of μ and σ 2 . Note: the density function of N( μ, σ 2 ) is f ( x ) = 1 2 πσ 2 e - ( x - μ ) 2 2 σ 2 . (2) Suppose X 1 , X 2 , ..., X n N( μ 1 , σ 2 ), and Y 1 , Y 2 , ..., Y m N( μ 2 , σ 2 ). Please find the MLE of μ 1 , μ 2 and σ 2 . 1. X 1 , · · · , X n N ( μ, σ 2 ) L ( μ, σ 2 ) = (2 πσ 2 ) - n 2 e - ( x i - μ ) 2 2 σ 2 l ( μ, σ 2 ) = - n 2 log 2 πσ 2 - ( x i - μ ) 2 2 σ 2 ∂l ∂μ = ( x i - μ ) σ 2 = 0 ∂l ∂σ 2 = - n 2 1 σ 2 + ( x i - μ ) 2 2 σ 4 = 0 ˆ μ MLE = ¯ X, ˆ σ 2 MLE = ( x i - ¯ X ) 2 n 2.
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