Definition (Multivariate Gaussian)

The column random vector $\underline X=(X_{1},\cdots,X_{n})^{T}$ with mean vector $\underline\mu = (\mu_{1},\cdots,\mu_{n})^{T}$, where $\mu_{i}=E[X_{i}], \ i=1,\cdots,n$, and covariance matrix $K_{\underline X}$ (assumed to be invertible) given by $$\begin{align*} K_{\underline X}&=E[(\underline X-\underline\mu)(\underline X-\underline\mu)^{T}] \end{align*}$$where the covariance — $\mbox{Cov}(X_{i},X_{j})=E[(x_{i}-\mu_{i})(x_{j}-\mu_{j})^{T}], \ i=1,\cdots n$ — is Gaussian if its joint pdf is given by: $$f_{\underline X}(\underline x)=\frac{1}{\left(\sqrt{2\pi}\right)^{n}\sqrt{\det(K_{\underline X})}}e^{-\frac{1}{2}(\underline x-\underline \mu)^{T}K_{\underline X}^{-1}(\underline x-\underline\mu)}, \ \underline x=(x_{1},\cdots,x_{n})^{T}\in\mathbb{R}^{n}$$