What of the generalization to a multivariate normal.
Let $(x^1, \dots , x^n)$ be a dataset in
$\R ^d$.
Let $f$ be a multivariate normal density with
mean
\[
\frac{1}{n} \sum_{k = 1}^{d} x^k
\] \[
\frac{1}{n}
\sum_{k = 1}^{n}
\left(x^k - \frac{1}{n} \sum_{k = 1}^{n} x^k\right)
\left(x^k - \frac{1}{n} \sum_{k = 1}^{n} x^k\right)^\tp
.
\]
We express the log likelihood
\[
\sum_{k = 1}^{n} -\frac{1}{2}(x - \mu )^\tp \Sigma ^{-1}
(x-\mu ) - \frac{1}{2}\log (2\pi )^d - \frac{1}{2} \log \det
\Sigma
\] \[
\frac{\partial \ell }{\partial P} = \sum_{k = 1}^{n} (x^k -
\mu )(x^k - \mu )^\tp - P^{-1}.
\]
We call these two objects the maximum likelihood mean or empirical mean and maximum likelihood covariance or empirical covariance of the dataset. We call the normal density with the empirical mean and empirical covariance the empirical normal of the dataset.