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Unformatted text preview: ∂a 0 = 2n2 a − 2n1 + 2n1a
2a(n2 + n1 ) = 2n1
a(n2 + n1 ) = n1
a= n1
n2 + n1 755
n −x ⎛ 1 ⎞ ∑θ n 2
L(θ ) = ⎜
∏ xi
⎜ 3⎟ e
⎟
⎝ 2θ ⎠
i =1
n i i =1 n
nx
⎛1⎞
i
ln L(θ ) = n ln⎜
⎜ 3 ⎟ + 2 ∑ ln xi − ∑ θ
⎟
⎝ 2θ ⎠ i =1
i =1 ∂ ln L(θ ) − 3n n xi
=
+∑
2
θ
∂θ
i =1θ
Making the last equation equal to zero and solving for theta, we obtain:
n ˆ
Θ= ∑ xi
i =1 3n as the maximum likelihood estimate. 756
n L(θ ) = θ n ∏ xiθ −1
i =1 n ln L(θ ) = n lnθ + (θ − 1) ∑ ln( xi )
i =1 ∂ ln L(θ ) n n
= + ∑ ln( xi )
∂θ
θ i =1 making the last equation equal to zero and solving for theta, we obtain the maximum likelihood estimate.
ˆ
Θ= −n
n ∑ ln( xi )
i =1 757 717 L(θ ) = 1 θ n n ∏...
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 Spring '14

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