Namespaces
Variants
Actions

Difference between revisions of "Marginal distribution"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (tex encoded by computer)
(latex details)
 
Line 37: Line 37:
  
 
$$  
 
$$  
F ( x _ {1} \dots x _ {n} )  = \  
+
F ( x _ {1} \dots x _ {n} )  = \prod_{i=1}^ { n } F _ {i} ( x _ {i} )
\prod _ { i= } 1 ^ { n }  
 
F _ {i} ( x _ {i} )
 
 
$$
 
$$
  
Line 45: Line 43:
  
 
$$  
 
$$  
p ( x _ {1} \dots x _ {n} )  = \  
+
p ( x _ {1} \dots x _ {n} )  = \prod_{i=1}^ { n } p _ {i} ( x _ {i} ) .
\prod _ { i= } 1 ^ { n }  
 
p _ {i} ( x _ {i} ) .
 
 
$$
 
$$
  
Line 53: Line 49:
  
 
====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>
+
<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>

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=55088
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article