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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.

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Definition

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]

Example

Let X be a $2\times 1$ random vector having joint probability mass function:[eq4]

In other words, the joint pdf is equal to 1 if both components of the vector belong to the interval $\left[ 0,1\right] $ 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:[eq5]

More details

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

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