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Marginal probability density function

Consider a random vector whose entries are continuous random variables, called a continuous random vector. When taken alone, one of the entries of the random vector has a univariate probability distribution that can be described by its probability density function. This is called marginal probability density function, in order to distinguish it from the joint probability density function, which instead describes the multivariate distribution of all the entries of the random vector taken together.

Table of Contents

Definition

A more formal definition follows.

Definition Let [eq1] be K continuous random variables forming a Kx1 random vector. Then, for each $i=1,ldots ,K$, the probability density function of the random variable X_i, denoted by [eq2], is called marginal probability density function.

Recall that the probability density function [eq3] is a function [eq4] such that, for any interval [eq5], we have[eq6]where [eq7] is the probability that X_i will take a value in the interval $left[ a,b
ight] $.

Instead, the joint probability density function of the vector X is a function [eq8] such that, for any hyper-rectangle[eq9]we have[eq10]where [eq11]is the probability that X_i will take a value in the interval [eq12], simultaneously for all $i=1,ldots ,K$.

How to derive it

The marginal probability density function of X_i is obtained from the joint probability density function as follows:[eq13]In other words, the marginal probability density function of X_i is obtained by integrating the joint probability density function with respect to all variables except $x_{i}$.

Example

Let X be a $2	imes 1$ absolutely continuous random vector having joint probability density function[eq14]The marginal probability density function of X_1 is obtained by integrating the joint probability density function with respect to $x_{2}$. When [eq15], then[eq16]When [eq17], then[eq18]Therefore, the marginal probability density function of X_1 is[eq19]

More details

Marginal probability density functions are discussed in more detail in the lecture entitled Random vectors.

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