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A general term referring to the distribution of several random variables defined on the same probability space. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054260/j0542601.png" /> be random variables defined on a [[Probability space|probability space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054260/j0542602.png" /> and taking values in measurable spaces (cf. [[Measurable space|Measurable space]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054260/j0542603.png" />. The joint distribution of these variables is the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054260/j0542604.png" /> of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054260/j0542605.png" />, defined by
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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
 +
 
 +
$$
 +
P _ {X _ {1}  \dots X _ {n} } ( B _ {1} \dots B _ {n} )  = \
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{\mathsf P} \{ X _ {1} \in B _ {1} \dots X _ {n} \in B _ {n} \} .
 +
$$
  
 
In connection with joint distributions, one speaks of the joint distribution function and of the joint probability density.
 
In connection with joint distributions, one speaks of the joint distribution function and of the joint probability density.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054260/j0542607.png" /> are ordinary real random variables, then their joint distribution is the distribution of the random vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054260/j0542608.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054260/j0542609.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054260/j05426010.png" /> (see [[Multi-dimensional distribution|Multi-dimensional distribution]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054260/j05426011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054260/j05426012.png" />, is a stochastic process, then the joint distributions of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054260/j05426013.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054260/j05426014.png" /> are called the finite-dimensional distributions of the stochastic process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054260/j05426015.png" />.
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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} ) $
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in $  n $-
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dimensional Euclidean space $  \mathbf R  ^ {n} $(
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see [[Multi-dimensional distribution|Multi-dimensional distribution]]). If $  X ( t) $,  
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$  t \in T $,  
 +
is a stochastic process, then the joint distributions of the variables $  X ( t _ {1} ) \dots X ( t _ {n} ) $
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for $  t _ {1} \dots t _ {n} \in T $
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are called the finite-dimensional distributions of the stochastic process $  X ( t) $.
  
 
====References====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 22:14, 5 June 2020


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)

Comments

References

[a1] P. Billingsley, "Probability and measure" , Wiley (1979)
[a2] J.L. Doob, "Stochastic processes" , Wiley (1953)
How to Cite This Entry:
Joint distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Joint_distribution&oldid=47467
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article