Gaussian Distribution

Let $X\in {\mathbb{R}}^n$ be a continuous random vector with the

Parameters

• $\mu \in {\mathbb{R}}^n$: the mean vector.
• The covariance matrix

$$\Sigma =\left(\begin{array}{cccc}\text{Var}(X_1)& \text{Cov}(X_1,X_2)& \cdots & \text{Cov}(X_1,X_n)\\ \text{Cov}(X_2,X_1)& \text{Var}(X_2)& \cdots & \text{Cov}(X_2,X_n)\\ ⋮& ⋮& \ddots & ⋮\\ \text{Cov}(X_n,X_1)& \cdots & \text{Cov}(X_n,X_{n-1})& \text{Var}(X_n)\end{array}\right).$$

The random variable is said to be Gaussian distributed if it described by the following

probability density function

$$\begin{array}{rrcl}f:& \mathbb{R}& \mapsto & \mathbb{R}\\ & x& \to & \frac{\exp \left(-\frac{(x-\mu {)}^T{\Sigma }^{-1}(x-\mu )}{2}\right)}{\sqrt{(2\pi {)}^k\det (\Sigma )}}.\end{array}$$