Difference between revisions of "Marginal distribution"
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− | The | + | 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} ) $ |
+ | 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 | 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. | The marginal distribution with respect to a probability distribution given on a product of spaces more general than real lines is defined similarly. |
Revision as of 07:59, 6 June 2020
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) |
Marginal distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Marginal_distribution&oldid=13898