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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 a continuous random vector.

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

The generalization works as follows:

Table of Contents


The following is a formal definition.

Definition Let X be a Kx1 continuous random vector. The joint probability density function of X is a function [eq1] such that[eq2]for any hyper-rectangle[eq3]

The notation[eq4]used in the definition above has the following meaning:

  1. the first entry of the vector X belongs to the interval [eq5];

  2. the second entry of the vector X belongs to the interval [eq6];

  3. and so on.

Furthermore, the notation[eq7]where x is a K-dimensional vector is used interchangeably with the notation[eq8]where [eq9] are the K entries of x.

Finally, the notation [eq10] means that the multiple integral is computed along all the K co-ordinates.


Here are some examples.

Example 1

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

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

Suppose we need to compute the probability that both components will be less than or equal to $1/2$. This probability can be computed as a double integral:[eq12]

Example 2

Let X be a $2	imes 1$ random vector having joint probability density function[eq13]

Suppose 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:[eq14]where in step $rame{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 entitled Random vectors.

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