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)