# 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:

${f}_{\{{X}_{i}:i\in I\}}(\mathbf{x})={\sum}_{\{{x}_{i}:i\notin I\}}{f}_{{X}_{1},\mathrm{\dots},{X}_{n}}({x}_{1},\mathrm{\dots},{x}_{n})$ or

${f}_{\{{X}_{i}:i\in I\}}(\mathbf{x})={\int}_{\{{x}_{i}:i\notin I\}}{f}_{{X}_{1},\mathrm{\dots},{X}_{n}}({u}_{1},\mathrm{\dots},{u}_{n}){\prod}_{\{{u}_{i}:i\notin I\}}d{u}_{i}$,

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 |