Obtain the maximum likelihood estimator of the mean vector and variance-covariance matrix of the multivariate normal distribution.

Let $\mathbf{X}_1, \mathbf{X}_2, \ldots, \mathbf{X}_n$ be a random sample from a multivariate normal distribution with mean vector $\boldsymbol{\mu} \in \mathbb{R}^p$ and variance-covariance matrix $\boldsymbol{\Sigma} \in \mathbb{R}^{p \times p}$, where $\boldsymbol{\Sigma}$ is symmetric and positive definite. The probability density function of the multivariate normal distribution is given by:

$$
f(\mathbf{x}) = \frac{1}{(2\pi)^{p/2} |\boldsymbol{\Sigma}|^{1/2}} \exp\left( -\frac{1}{2} (\mathbf{x} - \boldsymbol{\mu})^\top \boldsymbol{\Sigma}^{-1} (\mathbf{x} - \boldsymbol{\mu}) \right)
$$

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