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The distribution of a random variable, or set of random variables, obtained by considering a component, or subset of components, of a larger random vector (see [[Multi-dimensional distribution|Multi-dimensional distribution]]) with a given distribution. Thus the marginal distribution is the projection of the distribution of the random vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062340/m0623401.png" /> onto an axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062340/m0623402.png" /> or subspace defined by variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062340/m0623403.png" />, and is completely determined by the distribution of the original vector. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062340/m0623404.png" /> is the distribution function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062340/m0623405.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062340/m0623406.png" />, then the distribution function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062340/m0623407.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062340/m0623408.png" />; if the two-dimensional distribution is absolutely continuous and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062340/m0623409.png" /> is its density, then the density of the marginal distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062340/m06234010.png" /> is
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062340/m06234011.png" /></td> </tr></table>
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The marginal distribution is calculated similarly for any component or set of components of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062340/m06234012.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062340/m06234013.png" />. If the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062340/m06234014.png" /> is normal, then all marginal distributions are also normal. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062340/m06234015.png" /> are mutually independent, then the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062340/m06234016.png" /> is uniquely determined by the marginal distributions of the components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062340/m06234017.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062340/m06234018.png" />:
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The distribution of a random variable, or set of random variables, obtained by considering a component, or subset of components, of a larger random vector (see [[Multi-dimensional distribution|Multi-dimensional distribution]]) with a given distribution. Thus the marginal distribution is the projection of the distribution of the random vector $  X= ( X _ {1} \dots X _ {n} ) $
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onto an axis  $  x _ {1} $
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or subspace defined by variables  $  x _ {i _ {1}  } \dots x _ {i _ {k}  } $,
 +
and is completely determined by the distribution of the original vector. For example, if  $  F ( x _ {1} , x _ {2} ) $
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is the distribution function of $  X = ( X _ {1} , X _ {2} ) $
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in  $  \mathbf R  ^ {2} $,  
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then the distribution function of $  X _ {1} $
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is equal to  $  F _ {1} ( x _ {1} ) = F ( x _ {1} , + \infty ) $;
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if the two-dimensional distribution is absolutely continuous and if  $  p ( x _ {1} , x _ {2} ) $
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is its density, then the density of the marginal distribution of $  X _ {1} $
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is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062340/m06234019.png" /></td> </tr></table>
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$$
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p _ {1} ( x _ {1} )  = \
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\int\limits _ {- \infty } ^ { {+ }  \infty } p ( x _ {1} , x _ {2} )  d x _ {2} .
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$$
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 +
The marginal distribution is calculated similarly for any component or set of components of the vector  $  X = ( X _ {1} \dots X _ {n} ) $
 +
for any  $  n $.
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If the distribution of  $  X $
 +
is normal, then all marginal distributions are also normal. When  $  X _ {1} \dots X _ {n} $
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are mutually independent, then the distribution of  $  X $
 +
is uniquely determined by the marginal distributions of the components  $  X _ {1} \dots X _ {n} $
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of  $  X $:
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 +
$$
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F ( x _ {1} \dots x _ {n} )  = \prod_{i=1}^ { n } F _ {i} ( x _ {i} )
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$$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062340/m06234020.png" /></td> </tr></table>
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$$
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p ( x _ {1} \dots x _ {n} )  = \prod_{i=1}^ { n } p _ {i} ( x _ {i} ) .
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$$
  
 
The marginal distribution with respect to a probability distribution given on a product of spaces more general than real lines is defined similarly.
 
The marginal distribution with respect to a probability distribution given on a product of spaces more general than real lines is defined similarly.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Loève,  "Probability theory" , Springer  (1977)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Cramér,  "Mathematical methods of statistics" , Princeton Univ. Press  (1946)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  M. Loève,  "Probability theory" , Springer  (1977)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  H. Cramér,  "Mathematical methods of statistics" , Princeton Univ. Press  (1946)</TD></TR>
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</table>

Latest revision as of 07:40, 14 January 2024


The distribution of a random variable, or set of random variables, obtained by considering a component, or subset of components, of a larger random vector (see Multi-dimensional distribution) with a given distribution. Thus the marginal distribution is the projection of the distribution of the random vector $ X= ( X _ {1} \dots X _ {n} ) $ onto an axis $ x _ {1} $ or subspace defined by variables $ x _ {i _ {1} } \dots x _ {i _ {k} } $, and is completely determined by the distribution of the original vector. For example, if $ F ( x _ {1} , x _ {2} ) $ is the distribution function of $ X = ( X _ {1} , X _ {2} ) $ in $ \mathbf R ^ {2} $, then the distribution function of $ X _ {1} $ is equal to $ F _ {1} ( x _ {1} ) = F ( x _ {1} , + \infty ) $; if the two-dimensional distribution is absolutely continuous and if $ p ( x _ {1} , x _ {2} ) $ is its density, then the density of the marginal distribution of $ X _ {1} $ is

$$ p _ {1} ( x _ {1} ) = \ \int\limits _ {- \infty } ^ { {+ } \infty } p ( x _ {1} , x _ {2} ) d x _ {2} . $$

The marginal distribution is calculated similarly for any component or set of components of the vector $ X = ( X _ {1} \dots X _ {n} ) $ for any $ n $. If the distribution of $ X $ is normal, then all marginal distributions are also normal. When $ X _ {1} \dots X _ {n} $ are mutually independent, then the distribution of $ X $ is uniquely determined by the marginal distributions of the components $ X _ {1} \dots X _ {n} $ of $ X $:

$$ F ( x _ {1} \dots x _ {n} ) = \prod_{i=1}^ { n } F _ {i} ( x _ {i} ) $$

and

$$ p ( x _ {1} \dots x _ {n} ) = \prod_{i=1}^ { n } p _ {i} ( x _ {i} ) . $$

The marginal distribution with respect to a probability distribution given on a product of spaces more general than real lines is defined similarly.

References

[1] M. Loève, "Probability theory" , Springer (1977)
[2] H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946)
How to Cite This Entry:
Marginal distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Marginal_distribution&oldid=13898
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article