### MULTIVARIATE GAUSSIAN:

The univariate Gaussian ($$X \sim N(\mu, \sigma^2$$) is:

$f(x) = \frac{1}{\sqrt{2\pi\sigma^2}}\exp\left(-\frac{1}{2\sigma^2}(x-\mu)^2\right), \forall \in \mathbb R.$

The degenerate Gaussian has variance equal to $$0$$ and hence, $$X(\omega)=\mu, \forall \omega \in \Omega.$$

The multivariate Gaussian is defined for $$X \in \mathbb R^n$$ as any linear combination of univariate Gaussian distributions $$X_i$$:

$$a^T X = \sum_{i=1}^n a_i X_i$$ for $$\forall a \in \mathbb R^n.$$

We will express it as $$X\sim N(\mu,\Sigma)$$, where $$\mu$$ is a vector in $$\mathbb R^n$$. The $$\mathbb E(X_i)=\mu_i$$; and the covariance matrix is an $$n \times n$$ positive semidefinite matrix, such that $$\Sigma(X_i, X_j) = \text{Cov}_{ij}.$$

A multivariate Gaussian is degenerated if the $$\text{det}(\Sigma)=0.$$

If the Gaussian distributions (components) are independent, the covariance matrix is:

$\Sigma = \begin{bmatrix}\sigma_1^2&0\\0&\sigma_2^2\end{bmatrix}$

If the variance is $$1$$ for both Gaussians: