StatlectThe Digital Textbook

# Joint probability density function

The concept of joint probability density function (joint pdf) is a multivariate generalization of the concept of probability density function. The joint pdf characterizes the distribution of a continuous random vector. The probability that the realization of a continuous random vector will belong to a given set is equal to the multiple integral of the joint pdf over that set.

## Definition

The following is a formal definition.

Definition Let be a continuous random vector. The joint probability density function of is a function such thatfor any hyper-rectangle

## Example

Let be a random vector having joint probability mass function:

In other words, the joint pdf is equal to if both components of the vector belong to the interval and it is equal to otherwise.

Suppose we need to compute the probability that both components will be less than or equal to . This probability can be computed as a double integral:

## More details

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