Let (mathbf{X}) be an (n)-dimensional normal random vector with mean vector (boldsymbol{mu}) and covariance matrix (boldsymbol{Sigma}), where

Question:

Let $\mathbf{X}$ be an $n$-dimensional normal random vector with mean vector $\boldsymbol{\mu}$ and covariance matrix $\boldsymbol{\Sigma}$, where the determinant of $\boldsymbol{\Sigma}$ is non-zero. Show that $\boldsymbol{X}$ has joint probability density

\[ f_{X}(x)=\frac{1}{\sqrt{(2 \pi)^{n}|\Sigma|}} \mathrm{e}^{-\frac{1}{2}(x-\mu)^{\top} \Sigma^{-1}(x-\mu)}, \quad x \in \mathbb{R}^{n} \]

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