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STAT100B_HW6S

# STAT100B_HW6S - STAT 100B Homework 6 Solutions Denise Tsai...

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STAT 100B: Homework 6 Solutions Denise Tsai March 3, 2011 1. Suppose X 1 , X 2 , ..., X n Bernoulli( p ) independently. Please calculate the maximum likelihood estimate of p . Solution: Probability mass function: f ( x | p ) = p if x = 1 1 - p if x = 0 0 otherwise Likelihood function: L ( p ) = f ( x 1 , x 2 , · · · , x n | p ) = n Y i =1 p x i (1 - p ) 1 - x i = [ p x 1 (1 - p ) 1 - x 1 ][ p x 2 (1 - p ) 1 - x 2 ] · · · [ p x n (1 - p ) 1 - x n ] = p n i =1 x i (1 - p ) n - n i =1 x i Log-likelihood function: ( p ) = n X i =1 x i ln p + ( n - n X i =1 x i ) ln(1 - p ) Maximum likelihood estimate of p: 0 ( p ) = n i =1 x i p - n - n i =1 1 - p = 0 n X i =1 x i - p n X i =1 x i - np + p n X i =1 x i = 0 ˆ p = n i =1 x i n = ¯ x 1

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2. Suppose X 1 , X 2 , ..., X n N( μ, σ 2 ) independently. Please calculate the maximum likeli- hood estimate of ( μ, σ 2 ). Solution: Probability density function: f ( x | μ, σ 2 ) = 1 2 πσ 2 e - ( x - μ ) 2 2 σ 2 Likelihood function: L ( μ, σ 2 ) = f ( x 1 , x 2 , · · · , x n | μ, σ 2 ) = n Y i =1 1 2 πσ 2 e - ( x i - μ ) 2 2 σ 2 = (2 πσ 2 ) - n 2 e n i =1 - ( x i - μ ) 2 2 σ 2 Log-likelihood function: (
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STAT100B_HW6S - STAT 100B Homework 6 Solutions Denise Tsai...

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