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Marginal distribution function

Given a random vector, the probability distribution of all its components, considered together, is called joint distribution, while the probability distribution of one of its components, considered in isolation, is called marginal distribution.

Definition

The following is a more precise definition.

Definition Let X_i be the i-th component of a Kx1 random vector X having joint distribution function [eq1]. The distribution function of X_i is called marginal distribution function of X_i and it is denoted by [eq2].

Marginal distributions and independence

Marginal distribution functions play an important role in the characterization of independence between random variables: two random variables are independent if and only if their joint distribution function is equal to the product of their marginal distribution functions (see the lecture entitled Independent random variables).

Example Let X and Y be two random variables having marginal distribution functions[eq3]and joint distribution function[eq4]It is easy to check that [eq5]for any x and $y$, which implies that X and Y are independent.

More details

A more detailed discussion of the marginal distribution function can be found in the lecture entitled Random vectors.

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