Joint distribution
In the study of probability, given at least two random variables X, Y, ..., that are defined on a probability space, the joint probability distribution for X, Y, ... is a probability distribution that gives the probability that each of X, Y, ... 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.
Consider the flip of two fair coins; let A and B be discrete random variables associated with the outcomes first and second coin flips respectively. If a coin displays "heads" then associated random variable is 1, and is 0 otherwise. The joint probability density function of A and B defines probabilities for each pair of outcomes. All possible outcomes are
Since each outcome is equally likely the joint probability density function becomes
when . Since the coin flips are independent, the joint probability density function is the product of the marginals:
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