Definition:Joint Distribution

Definition

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\struct {S, \Sigma'}$ be a measurable space.

Let $X_1, X_2, \ldots, X_d$ be random variables on $\struct {\Omega, \Sigma, \Pr}$ taking values in $\struct {S, \Sigma'}$.

Define a random vector $X : \Omega \to S^d$ by:

$\map X \omega = \tuple {\map {X_1} \omega, \map {X_2} \omega, \ldots, \map {X_d} \omega}$

for each $\omega \in \Omega$.

Then the joint distribution of $X_1, X_2, \ldots, X_d$ is defined as the probability distribution of $X$.

Also known as

A joint distribution is also known as a multivariate distribution.

When the number of random variables is $2$, the term bivariate distribution is usually used.

Also see

• Results about joint distributions can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): joint distribution
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): multivariate distribution
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): joint distribution
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): multivariate distribution
• 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $10$: Probability: $10.1$: Basics