ProBaBility, verBal exPreSSionS

Axiomatically, the definition of the joint prob­ ability of several continuous variables is based on a cumulative distribution function, F(x, y, z, K), whose properties are analogous to the case of a single variable, and the probability density function derives from it. However, in practice, the probability P is never defined from a cumulative function, F, but from a density function, f, assum­ ing that f belongs to one of the families of multi­ variate continuous probability distributions, such as the multivariate Gaussian, also called multivari­ ate normal. Other families of continuous probabil­ ity density distributions can be found in statistical textbooks and on the Internet. The derivation of the marginal density function from a joint probability density is analogous to the case of discrete variables, just replacing the sum with an integral. For example, in the case of a joint density f(x, y) defined over two variables X and Y, the marginal density distributions are