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Marginal probability density function

by , PhD

Consider a continuous random vector, whose entries are continuous random variables.

Each entry of the random vector has a univariate distribution described by a probability density function (pdf). This is called marginal probability density function, to distinguish it from the joint probability density function, which depicts the multivariate distribution of all the entries of the random vector.

Table of Contents

Definition

A more formal definition follows.

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

What you need to know

Before explaining how to derive the marginal pdfs from the joint pdf, let us revise the basics of pdfs.

Pdf

The probability density function of a variable X_i is a function [eq3] such that the probability that X_i will take a value in the interval $left[ a,b
ight] $ is[eq4]for any interval [eq5]

Joint pdf

The joint probability density function of the vector X is a function [eq6] such that the probability that X_i will take a value in the interval [eq7], simultaneously for all $i=1,ldots ,K$, is[eq8]for any hyper-rectangle[eq9]

How to derive the marginal pdf

The marginal probability density function of X_i is obtained from the joint pdf as follows:[eq10]

In other words, to compute the marginal pdf of X_i, we integrate the joint pdf with respect to all the variables except $x_{i}$.

Example

Let X be a $2	imes 1$ continuous random vector having joint pdf[eq11]

To derive the marginal probability density function of X_1, we integrate the joint pdf with respect to $x_{2}$.

When [eq12], then[eq13]

When [eq14], then[eq15]

Therefore, the marginal probability density function of X_1 is[eq16]

Other examples

On the following pages you can find other examples and detailed derivations:

More details

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

Keep reading the glossary

Previous entry: Marginal distribution function

Next entry: Marginal probability mass function

How to cite

Please cite as:

Taboga, Marco (2021). "Marginal probability density function", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/marginal-probability-density-function.

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