# Marginal distribution

The distribution of a random variable, or set of random variables, obtained by considering a component, or subset of components, of a larger random vector (see Multi-dimensional distribution) with a given distribution. Thus the marginal distribution is the projection of the distribution of the random vector onto an axis or subspace defined by variables , and is completely determined by the distribution of the original vector. For example, if is the distribution function of in , then the distribution function of is equal to ; if the two-dimensional distribution is absolutely continuous and if is its density, then the density of the marginal distribution of is

The marginal distribution is calculated similarly for any component or set of components of the vector for any . If the distribution of is normal, then all marginal distributions are also normal. When are mutually independent, then the distribution of is uniquely determined by the marginal distributions of the components of :

and

The marginal distribution with respect to a probability distribution given on a product of spaces more general than real lines is defined similarly.

#### References

[1] | M. Loève, "Probability theory" , Springer (1977) |

[2] | H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) |

**How to Cite This Entry:**

Marginal distribution.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Marginal_distribution&oldid=13898