Introduction to Probability and Statistics for Engineers and Scientists

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