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: