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marginal distribution (Algorithm)

Given random variables $ X_1, X_2, ... , X_n$ and a subset $ I \subset \{1,2,...,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,...,X_n}(x_1,...,x_n) }$ or
$ f_{\{X_i : i \in I\}}(\mathbf{x}) = \int_{\{x_i : i \notin I\}}^{}{ f_{X_1,...,X_n}(u_1,...,u_n) \prod_{ \{u_i : i \notin I\} }^{} du_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,...,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).



"marginal distribution" is owned by mathcam. [ full author list (2) | owner history (1) ]
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Other names:  marginal density function, marginal probability function
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Cross-references: joint distribution, continuous, discrete, variables, summing, subset, random variables
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This is version 5 of marginal distribution, born on 2001-10-26, modified 2004-06-09.
Object id is 591, canonical name is MarginalDistribution.
Accessed 21473 times total.

Classification:
AMS MSC: 60E05 (Probability theory and stochastic processes :: Distribution theory :: Distributions: general theory)
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