236 Chapter 7: Parameter Estimation
EXAMPLE 7.2e Maximum Likelihood Estimator in a Normal Population Suppose X 1 ,...,Xn
are independent, normal random variables each with unknown meanμand unknown
standard deviationσ. The joint density is given by
f(x 1 ,...,xn|μ,σ)=
∏n
i= 1
1
√
2 πσ
exp
[
−(xi−μ)^2
2 σ^2
]
=
(
1
2 π
)n/2
1
σn
exp
−
∑n
1
(xi−μ)^2
2 σ^2
The logarithm of the likelihood is thus given by
logf(x 1 ,...,xn|μ,σ)=−
n
2
log(2π)−nlogσ−
∑n
1
(xi−μ)^2
2 σ^2
In order to find the value ofμandσmaximizing the foregoing, we compute
∂
∂μ
logf(x 1 ,...,xn|μ,σ)=
∑n
i= 1
(xi−μ)
σ^2
∂
∂σ
logf(x 1 ,...,xn|μ,σ)=−
n
σ
+
∑n
1
(xi−μ)^2
σ^3
Equating these equations to zero yields that
μˆ=
∑n
i= 1
xi/n
and
σˆ =
[ n
∑
i= 1
(xi−ˆμ)^2 /n
]1/2