Given random variables ${\displaystyle X,Y,\ldots }$, that are defined on the same probability space, the multivariate or 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.

The joint probability distribution can be expressed in terms of a joint cumulative distribution function and either 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.

Each of two urns contains twice as many red balls as blue balls, and no others, and one ball is randomly selected from each urn, with the two draws independent of each other. Let ${\displaystyle A}$ and ${\displaystyle B}$ be discrete random variables associated with the outcomes of the draw from the first urn and second urn respectively. The probability of drawing a red ball from either of the urns is ⁠2/3⁠, and the probability of drawing a blue ball is ⁠1/3⁠. The joint probability distribution is presented in the following table:

Each of the four inner cells shows the probability of a particular combination of results from the two draws; these probabilities are the joint distribution. In any one cell the probability of a particular combination occurring is (since the draws are independent) the product of the probability of the specified result for A and the probability of the specified result for B. The probabilities in these four cells sum to 1, as with all probability distributions.

Moreover, the final row and the final column give the marginal probability distribution for A and the marginal probability distribution for B respectively. For example, for A the first of these cells gives the sum of the probabilities for A being red, regardless of which possibility for B in the column above the cell occurs, as ⁠2/3⁠. Thus the marginal probability distribution for ${\displaystyle A}$ gives ${\displaystyle A}$'s probabilities unconditional on ${\displaystyle B}$, in a margin of the table.

Consider the flip of two fair coins; let ${\displaystyle A}$ and ${\displaystyle B}$ be discrete random variables associated with the outcomes of the first and second coin flips respectively. Each coin flip is a Bernoulli trial and has a Bernoulli distribution. If a coin displays "heads" then the associated random variable takes the value 1, and it takes the value 0 otherwise. The probability of each of these outcomes is ⁠1/2⁠, so the marginal (unconditional) density functions are

$${\displaystyle {\begin{aligned}P(A)&=1/2\quad {\text{for}}\quad A\in \{0,1\};\\P(B)&=1/2\quad {\text{for}}\quad B\in \{0,1\}.\end{aligned}}}$$

The joint probability mass function of ${\displaystyle A}$ and ${\displaystyle B}$ defines probabilities for each pair of outcomes. All possible outcomes are $${\displaystyle (A,B)\in \{(0,0),(0,1),(1,0),(1,1)\}.}$$ Since each outcome is equally likely the joint probability mass function becomes $${\displaystyle P(A,B)=1/4\quad {\text{for}}\quad A,B\in \{0,1\}.}$$