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100BHWS2

# 100BHWS2 - ,θ x i = 1 θ n I θ ≥ x max ˆ θ = x max...

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STAT 100B HW2 Solution Problem 1 Bernoulli f ( x ) = p x (1 - p ) 1 - x , x ∈ { 0 , 1 } L ( p ) = Q n i =1 f ( x i ) = p x i (1 - p ) n - x i l ( p ) = x i ln p + ( n - x i ) ln(1 - p ) ∂l ( p ) ∂p = x i p - n - x i 1 - p = 0 x i (1 - p ) = np - p x i ˆ p = ¯ x Problem 2 Normal f ( x ) = 1 2 πσ 2 e - ( x - μ ) 2 2 σ 2 , x R L ( μ, σ 2 ) = Q n i =1 f ( x i ) = (2 πσ 2 ) - n 2 e - ( x i - μ ) 2 2 σ 2 l ( μ, σ 2 ) = - n 2 ln(2 πσ 2 ) - ( x i - μ ) 2 2 σ 2 ∂l ( μ,σ 2 ) ∂μ = ( x i - μ ) 2 2 σ 2 = 0 ˆ μ = ¯ x ∂l ( μ,σ 2 ) ∂σ 2 = - n 2 σ 2 + ( x i - μ ) 2 2 σ 4 = 0 ˆ σ 2 = ( x i - ˆ μ ) 2 n = ( x i - ¯ x ) 2 n Problem 3 Poisson f ( x ) = e - λ λ x x ! , x ∈ { 0 , 1 , 2 , · · ·} L ( λ ) = e - λ x i Q x i ! l ( λ ) = - + x i ln λ - ln( Q x i !) ∂l ( λ ) ∂λ = - n + x i λ = 0 ˆ λ = ¯ x Problem 4 Exponential f ( x ) = λe - λx , x 0 L ( λ ) = λ n e - λ x i l ( λ ) = n ln λ - λ x i ( λ ) ∂λ = n λ - x i = 0 ˆ λ = 1 ¯ x Problem 5 Geometric f ( x ) = (1 - p ) x - 1 p, x ∈ { 1 , 2 , 3 , · · ·} L ( p ) = (1 - p ) x i - n p n l ( p ) = ( x i - n ) ln(1 - p ) + n ln p ∂l ( p ) ∂p = ( x i - n )( - 1) 1 - p + n p = 0 ˆ p = 1 ¯ x Problem 6 Uniform f ( x ) = 1 θ , x
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Unformatted text preview: ,θ ] ( x i ) = 1 θ n I ( θ ≥ x max ) ˆ θ = x max Problem 7 f ( y ) = 1 √ 2 πσ 2 e-( y-βx ) 2 2 σ 2 L ( β,σ 2 ) = (2 πσ 2 )-n 2 e-∑ ( y i-βx i ) 2 2 σ 2 l ( β,σ 2 ) =-n 2 ln(2 πσ 2 )-∑ ( y i-βx i ) 2 2 σ 2 ∂l ( β,σ 2 ) ∂β = 2 ∑ ( y i-βx i ) x i 2 σ 2 = 0 ⇒ ˆ β = ∑ x i y i ∑ x 2 i ∂l ( β,σ 2 ) ∂σ 2 =-n 2 σ 2 + ∑ ( y i-βx i ) 2 2 σ 4 = 0 ˆ σ 2 = ∑ ( y i-ˆ βx i ) 2 n...
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