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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|probability space]]  $  \{ \Omega , {\mathcal A} , {\mathsf P} \} $
and taking values in measurable spaces (cf. [[Measurable space|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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.V. [Yu.V. Prokhorov] Prohorov,   Yu.A. Rozanov,   "Probability theory, basic concepts. Limit theorems, random processes" , Springer  (1969)  (Translated from Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Billingsley,   "Probability and measure" , Wiley  (1979)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Doob,   "Stochastic processes" , Wiley  (1953)</TD></TR></table>