Maximum Likelihood Estimation of Normal
Distribution
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The probability density (likelihood) of a single variable
drawn from a Normal distribution 
is:
Thus the likelihood of 
independent
draws 
from
this distribution is:
To identify the values of 
and 
that maximise 
it is easiest to
identify the maximum of the log likelihood (as the two will be maximised for
the same values of and since 
is
a monotonically increasing function of 
).
The log likelihood is:
This is maximised when 
and
,
i.e. when:
The maximum likelihood estimator, 
,
of the mean is thus the average of the
observations, 
.
It is possible to show that this is also the minimum variance unbiased
estimator of . The maximum likelihood estimator,
,
of is
the population standard deviation, 
of
the 
which
can be determined using the MnPopulationStdev
web function. Please note that whilst is
an unbiased estimator of , is
a biased estimator of . The minimum variance unbiased
estimator of 
is
the sample variance (i.e. square of the sample standard
deviation). The sample standard deviation is:
