👑 Gaussian

The Gaussian distribution, also called the normal distribution, is the most common distribution for continuous 🪐 Random Variables. The probability density is given by

In the multivariate case,

Rather than writing the whole formula, we commonly denote

Note

With and , we have the standard normal distribution.

Marginals and Conditionals §

Consider two multivariate random variables and with

The conditional distribution is a Gaussian,

where and .

The marginal distribution is also Gaussian,

Sums and Transformations §

The sum of two independent Gaussian random variables with and is a Gaussian given by

Moreover, if and are scaled, we have

Furthermore, if we transform using 🗺️ Linear Mapping so that , if ,