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Joint probability density function

by , PhD

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.

Table of Contents


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

The main differences between the pdf of a single variable and the joint pdf of multiple variables.

The generalization works as follows:

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 X be a continuous random vector formed by the random variables [eq1]. The joint probability density function of X is a function [eq2] such that[eq3]for any choice of the intervals[eq4]

Note that[eq5]is the probability that the following conditions are simultaneously satisfied:

  1. the first entry of the vector X belongs to the interval [eq6];

  2. the second entry of the vector X belongs to the interval [eq7];

  3. and so on.

The notation [eq8] means that the multiple integral is computed along all the K coordinates (see below for more details).

We often denote the joint pdf by[eq9]which is equivalent to [eq10]where [eq11] are the K entries of the vector x.

How to work out the multiple integral

Multiple integrals are relatively easy to work out.

Take, for example, a double integral:[eq12]where $x_{1}$ takes values in the interval [eq13] and $x_{2}$ in the interval [eq14].

The integral is computed in two steps:

  1. in the first step, we compute the inner integral[eq15]which gives a function of $x_{1}$ ($x_{2}$ disappears because it has been integrated out);

  2. in the second step, we compute the integral of [eq16], that is,[eq17]

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.

Example 1

Consider the joint pdf of two variables[eq18]

In other words, the joint pdf is equal to 1 if both entries of the vector belong to the interval $left[ 0,1
ight] $ and it is equal to 0 otherwise.

Suppose that we need to compute the probability that both entries will be less than or equal to $1/2$.

This probability can be computed as a double integral:[eq19]

Example 2

Consider two variables having joint probability density function[eq20]

Suppose that we want to calculate the probability that X_1 is greater than or equal to $frac{3}{2}$ and at the same time X_2 is less than or equal to 1.

This can be accomplished as follows:[eq21]where in step $frame{A}$ we have performed an integration by parts.

Joint and marginal density

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.

More details

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

Keep reading the glossary

Previous entry: Joint distribution function

Next entry: Joint probability mass function

How to cite

Please cite as:

Taboga, Marco (2021). "Joint probability density function", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix.

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