joint probability

${\displaystyle X}$

${\displaystyle Y}$

${\displaystyle p(X)}$

${\displaystyle p(Y)}$

Many sample observations (black) are shown from a joint probability distribution. The marginal densities are shown as well.

Given random variables ${\displaystyle X,Y,\ldots }$, that are defined on a probability space, the joint probability distribution for ${\displaystyle X,Y,\ldots }$ is a probability distribution that gives the probability that each of ${\displaystyle X,Y,\ldots }$ falls in any particular range or discrete set of values specified for that variable. In the case of only two random variables, this is called a bivariate distribution, but the concept generalizes to any number of random variables, giving a multivariate distribution.

The joint probability distribution can be expressed either in terms of a joint cumulative distribution function or in terms of a joint probability density function (in the case of continuous variables) or joint probability mass function (in the case of discrete variables). These in turn can be used to find two other types of distributions: the marginal distribution giving the probabilities for any one of the variables with no reference to any specific ranges of values for the other variables, and the conditional probability distribution giving the probabilities for any subset of the variables conditional on particular values of the remaining variables.