Marginal distribution

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)