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)

References

[a1]	P. Billingsley, "Probability and measure" , Wiley (1979)
[a2]	J.L. Doob, "Stochastic processes" , Wiley (1953)