marginal distribution

Given random variables $X_1,X_2,\mathrm{\dots },X_n$ and a subset $I\subset \{1,2,\mathrm{\dots },n\}$, the marginal distribution of the random variables $X_i:i\in I$ is the following:

$f_{\{X_i:i\in I\}}(𝐱)={\sum }_{\{x_i:i\notin I\}}{f}_{X_1,\mathrm{\dots },X_n}(x_1,\mathrm{\dots },x_n)$ or
$f_{\{X_i:i\in I\}}(𝐱)={\int }_{\{x_i:i\notin I\}}{f}_{X_1,\mathrm{\dots },X_n}(u_1,\mathrm{\dots },u_n){\prod }_{\{u_i:i\notin I\}}d{u}_i$,

summing if the variables are discrete and integrating if the variables are continuous.

This is, the marginal distribution of a set of random variables $X_1,\mathrm{\dots },X_n$ can be obtained by summing (or integrating) the joint distribution over all values of the other variables.

The most common marginal distribution is the individual marginal distribution (ie, the marginal distribution of ONE random variable).

Title	marginal distribution
Canonical name	MarginalDistribution
Date of creation	2013-03-22 11:55:00
Last modified on	2013-03-22 11:55:00
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	10
Author	mathcam (2727)
Entry type	Algorithm
Classification	msc 60E05
Synonym	marginal density function
Synonym	marginal probability function