Joint probability distribution

${\displaystyle X}$

${\displaystyle Y}$

${\displaystyle p(X)}$

${\displaystyle p(Y)}$

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

Part of a series on statistics
Probability theory
Probability Axioms Determinism System Indeterminism Randomness
Probability space Sample space Event Collectively exhaustive events Elementary event Mutual exclusivity Outcome Singleton Experiment Bernoulli trial Probability distribution Bernoulli distribution Binomial distribution Exponential distribution Normal distribution Pareto distribution Poisson distribution Probability measure Random variable Bernoulli process Continuous or discrete Expected value Variance Markov chain Observed value Random walk Stochastic process
Complementary event Joint probability Marginal probability Conditional probability
Independence Conditional independence Law of total probability Law of large numbers Bayes' theorem Boole's inequality
Venn diagram Tree diagram
v t e

Given random variables ${\displaystyle X,Y,\ldots }$, that are defined on the same^[1] 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.