Multivariate Gaussian PDF

Assume $X$ is an independent Random Vector. Then theJoint Density is given by

$$f_{\mathbf{X}}(\mathbf{x})=f_{X_1}\left(x_1\right)\cdots f_{X_n}\left(x_n\right)$$

In the multivariate Gaussian Case, $X$ is normally distributed (Mean is in the center). Then we can represent $X$ as

$$X\sim \mathcal{N}({\mu }_X,C_X)$$

Where Mean Vector ${\mu }_X=\mathbb{E}[X]$ and Covariance Matrix $C_X=\text{C}[X,X]$

The PDF of the Multivariate Gaussian is then given by

$$f_{\mathbf{X}}(\mathbf{x})=\frac{1}{\sqrt{(2\pi {)}^n\det \left({\mathbf{C}}_{\mathbf{X}}\right)}}{\mathrm{e}}^{-\frac{1}{2}{\left(\mathbf{x}-{\mathbf{\mu }}_{\mathbf{X}}\right)}^{\top }{\mathbf{C}}_{\mathbf{X}}^{-1}\left(\mathbf{x}-{\mathbf{\mu }}_{\mathbf{X}}\right)}$$

Example for Random Vector of size 2

Here, $\mathcal{N}(0,1)\to$ Normal distribution with Mean $0$ and Variance $1$