Marginal Densities

Definition

The $i$th marginal density of a multivariate density is the density obtained by integrating over every component with a particular component fixed.

Similalry the $i,j$th marginal density of a multivariate density is the density obtained by integrated over every component with the $i$ and $j$th components fixed.

Notation

Let $f: \R ^d \to \R $ be a density. For $i = 1, \dots , d$, let $f_i: \R \to \R $ be defined by

\[ f(\xi ) = \int _{\Set*{x \in \R ^d}{x_i = \xi }} f \]