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100BHWVIS

# 100BHWVIS - STAT 100B HWVI Solution Problem 1 Suppose X1 X2...

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STAT 100B HWVI Solution Problem 1: Suppose X 1 , X 2 , ..., X n Bernoulli( p ) independently. Please calculate the maximum likelihood estimate of p . A: The likelihood is L ( p ) = Q n i =1 p X i (1 - p ) 1 - X i = p n i =1 X i (1 - p ) n - n i =1 X i . The log-likelihood is l ( p ) = log L ( p ) = n i =1 X i log p +( n - n i =1 X i ) log(1 - p ). By setting l 0 ( p ) = 0, we get n i =1 X i /p - ( n - n i =1 X i ) / (1 - p ) = 0. So ˆ p = n i =1 X i /n . Problem 2: Suppose X 1 , X 2 , ..., X n N( μ, σ 2 ) independently. Please calculate the maximum likelihood estimate of ( μ, σ 2 ). A: The likelihood is L ( μ, σ 2 ) = n Y i =1 ( 1 2 πσ 2 exp {- ( X i - μ ) 2 2 σ 2 } ) = 1 (2 πσ 2 ) n/ 2 exp {- 1 2 σ 2 n X i =1 ( X i - μ ) 2 } . The log-likelihood is l ( μ, σ 2 ) = - n 2 log(2 πσ 2 ) - 1 2 σ 2 n X i =1 ( X i - μ ) 2 . By setting ∂l ( μ, σ 2 ) /∂μ = 0, we have n i =1 ( X i - μ ) = 0, so ˆ μ = ¯ X . Let τ = σ 2 . By setting ∂l ( μ, τ ) /∂τ = 0, we have - n/τ + n i =1 ( X i - μ ) 2 2 = 0, so ˆ σ 2 = n i =1 ( X i - ˆ μ ) 2 /n . Problem 3: Suppose X 1 , X 2 , ..., X n Exponential( λ ) independently. Please calculate the maxi- mum likelihood estimate of

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100BHWVIS - STAT 100B HWVI Solution Problem 1 Suppose X1 X2...

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