Marginal probability density function

by Marco Taboga, 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.

Definition

A more formal definition follows.

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 is a function such that the probability that will take a value in the interval isfor any interval

Joint pdf

The joint probability density function of the vector is a function such that the probability that will take a value in the interval , simultaneously for all , isfor any hyper-rectangle

How to derive the marginal pdf

The marginal probability density function of is obtained from the joint pdf as follows:

In other words, to compute the marginal pdf of , we integrate the joint pdf with respect to all the variables except .

Example

Let be a continuous random vector having joint pdf

To derive the marginal probability density function of , we integrate the joint pdf with respect to .

When , then

When , then

Therefore, the marginal probability density function of is

Other examples

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

• the marginals of a multivariate normal density are univariate normals;

• the marginals of a multivariate Student density are univariate t;

• the marginals of a Dirichlet density are Beta pdfs.

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.