The joint probability density function (joint pdf) is a function used to characterize the probability distribution of several continuous random variables, which together form a continuous random vector.

It is a multivariate generalization of the probability density function (pdf), which describes the distribution of a single continuous variable.

The generalization works as follows:

the integral of the density of a continuous variable over an interval is equal to the probability that the variable will belong to that interval;

the multiple integral of the joint density of a continuous vector over a given set is equal to the probability that the random vector will belong to that set.

Do not worry if you do not know how to compute multiple integrals. We will explain them below!

The following is a formal definition.

Definition

Let

be a continuous random vector formed by the random variables

.

The joint probability density function of

is a function

such

thatfor

any choice of the

intervals

Note

thatis

the probability that the following conditions are

simultaneously satisfied:

the first entry of the vector

belongs to the interval

;

the second entry of the vector

belongs to the interval

;

and so on.

The notation

means that the multiple integral is computed along all the

coordinates (see below for more details).

We often denote the joint pdf

bywhich

is equivalent to

where

are the

entries of the vector

.

Multiple integrals are relatively easy to work out.

Take, for example, a double

integral:where

takes values in the interval

and

in the interval

.

The integral is computed in two steps:

in the first step, we compute the inner

integralwhich

gives a function of

(

disappears because it has been integrated out);

in the second step, we compute the integral of

,

that

is,

If there are more than two integrals, the procedure is similar: we work out the integrals one by one, starting from the innermost one.

Here are some examples.

Consider the joint pdf of two

variables

In other words, the joint pdf is equal to

if both entries of the vector belong to the interval

and it is equal to

otherwise.

Suppose that we need to compute the probability that both entries will be less

than or equal to

.

This probability can be computed as a double

integral:

Consider two variables having joint probability density

function

Suppose that we want to calculate the probability that

is greater than or equal to

and at the same time

is less than or equal to

.

This can be accomplished as

follows:where

in step

we have performed an integration by parts.

One of the entries of a continuous random vector, when considered in isolation, can be described by its probability density function, which is called marginal density.

The joint density can be used to derive the marginal density. How to do this is explained in the glossary entry about the marginal density function.

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

Previous entry: Joint distribution function

Next entry: Joint probability mass function

Please cite as:

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

Most of the learning materials found on this website are now available in a traditional textbook format.