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STAT100B_HW7S

STAT100B_HW7S - STAT 100B Homework 7 Solutions Denise Tsai...

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Unformatted text preview: STAT 100B: Homework 7 Solutions Denise Tsai March 14, 2011 1. Let Y 1 ,...,Y n ∼ Bernoulli( λ ) independently. Let [ X i | Y i = 1] ∼ N ( μ 1 ,σ 2 ) and [ X i | Y i = 0] ∼ N ( μ ,σ 2 ). (a) Write down the likelihood L ( θ ), where θ = ( λ,μ ,μ 1 ,σ 2 ). Solution: L ( θ ) = n Y i =1 P ( x i ,y i ) = n Y i =1 P ( x i | y i ) P ( y i ) = n Y i =1 f 1 ( x i ) y i f ( x i ) 1- y i λ y i (1- λ ) 1- y i = n Y i =1 [ λf 1 ( x i )] y i [(1- λ ) f ( x i )] 1- y i = n Y i =1 λ · 1 √ 2 πσ 2 e- ( x i- μ 1 ) 2 2 σ 2 y i (1- λ ) · 1 √ 2 πσ 2 e- ( x i- μ 1 ) 2 2 σ 2 1- y i = λ ∑ n i =1 y i (1- λ ) n- ∑ n i =1 y i 1 √ 2 πσ 2 n · exp- ∑ n i =1 ( x i- μ 1 ) 2 y i 2 σ 2- ∑ n i =1 ( x i- μ ) 2 (1- y i ) 2 σ 2 = λ ∑ n i =1 y i (1- λ ) n- ∑ n i =1 y i (2 πσ 2 )- n/ 2 · exp (- 1 2 σ 2 n X i =1 ( x i- μ 1 ) 2 y i + ( x i- μ ) 2 (1- y i ) ) 1 (b) Find the MLE of θ , by setting the derivatives of the log-likelihood equal to 0....
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STAT100B_HW7S - STAT 100B Homework 7 Solutions Denise Tsai...

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