The Multivariate Normal Distribution

Let $n$ be a positive integer, let $Z_1,…,Z_n$ be independent, standand normal random variables, and set $\textbf{Z}=[Z_1,…,Z_n]^T$. Then $\mathbf{Z}$ has mean vector $\mathbf{0}$ and variance-covariance matrix $\mathbf{I}$. The density function of $\mathbf{Z}$ is given by

$$\begin{equation} \tag{1} \label{f_z} f_\mathbf{Z}(\mathbf{z})=\frac{1}{(2\pi)^{n/2}}\exp(-\frac{1}{2}\mathbf{z}^T\mathbf{z}), \mathbf{z} \in \mathbb{R}^n. \end{equation}$$

Let $\mu \in \mathbb{R}^n$, let $\mathbf{B}$ be an invertible $n\times n$ matrix, and set $\mathbf{Y}=\mu+\mathbf{BZ}$. Then $\mathbf{Y}$ has mean vector $E\mathbf{Y}=E(\mu+\mathbf{BZ})=\mu+\mathbf{B}(E\mathbf{Z})=\mu$ and variance-covariance matrix

$$\tag{2} \Sigma=\mathrm{VC}(\mu+\mathbf{BZ})=\mathbf{B}[\mathrm{VC}(\mathbf{Z})]\mathbf{B}^T=\mathbf{BIB}^T=\mathbf{BB}^T.$$

Hence, the density function of $\mathbf{Y}$ is given by

$$\tag{3} f_\mathbf{Y}(\mathbf{y})=\frac{1}{|\det{\mathbf{B}}|}f_{\mathbf{Z}}(\mathbf{B}^{-1}(\mathbf{y}-\mu)), \mathbf{y} \in \mathbb{R}^n.$$

Now $\det(\mathbf{\Sigma})=\det(\mathbf{BB^T})=[\det(\mathbf{B})]^2>0$ and hence $|\det(\mathbf{B})|=\sqrt{\det(\mathbf{\Sigma})}$. And according to $\eqref{f_z}$,

$$f_\mathbf{Z}(\mathbf{B}^{-1}(\mathbf{y}-\mu))=\frac{1}{(2\pi)^{n/2}}\exp\{-\frac{1}{2}[\mathbf{B}^{-1}(\mathbf{y}-\mu)]^T[\mathbf{B}^{-1}(\mathbf{y}-\mu)]\}.$$

Observe that $[\mathbf{B}^{-1}(\mathbf{y}-\mu)]^T[\mathbf{B}^{-1}(\mathbf{y}-\mu)]=(\mathbf{y}-\mu)^T\mathbf{\Sigma^{-1}}(\mathbf{y}-\mu)$, and hence the density function of $\mathbf{Y}$ is given by

$$\tag{4} f_\mathbf{Z}(\mathbf{B}^{-1}(\mathbf{y}-\mu))=\frac{1}{(2\pi)^{n/2}}\exp\{-\frac{1}{2}(\mathbf{y}-\mu)^T\mathbf{\Sigma^{-1}}(\mathbf{y}-\mu)\},\mathbf{y}\in\mathbb{R}^n.$$