# Joint distribution

A general term referring to the distribution of several random variables defined on the same probability space. Let $X _ {1} \dots X _ {n}$ be random variables defined on a probability space $\{ \Omega , {\mathcal A} , {\mathsf P} \}$ and taking values in measurable spaces (cf. Measurable space) $( \mathfrak X _ {k} , \mathfrak B _ {k} )$. The joint distribution of these variables is the function $P _ {X _ {1} \dots X _ {n} } ( B _ {1} \dots B _ {n} )$ of sets $B _ {1} \in {\mathcal B} _ {1} \dots B _ {n} \in {\mathcal B} _ {n}$, defined by

$$P _ {X _ {1} \dots X _ {n} } ( B _ {1} \dots B _ {n} ) = \ {\mathsf P} \{ X _ {1} \in B _ {1} \dots X _ {n} \in B _ {n} \} .$$

In connection with joint distributions, one speaks of the joint distribution function and of the joint probability density.

If $X _ {1} \dots X _ {n}$ are ordinary real random variables, then their joint distribution is the distribution of the random vector $( X _ {1} \dots X _ {n} )$ in $n$- dimensional Euclidean space $\mathbf R ^ {n}$( see Multi-dimensional distribution). If $X ( t)$, $t \in T$, is a stochastic process, then the joint distributions of the variables $X ( t _ {1} ) \dots X ( t _ {n} )$ for $t _ {1} \dots t _ {n} \in T$ are called the finite-dimensional distributions of the stochastic process $X ( t)$.

#### References

 [1] Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian)