Given a random vector, the probability distribution of all its components, considered together, is called joint distribution, while the probability distribution of one of its components, considered in isolation, is called marginal distribution.
The following is a more precise definition.
Definition
Let
be the
-th
component of a
random vector
having joint distribution
function
![[eq1]](/images/marginal_distribution_function__5.png){kind=small}
.
The distribution function of
is called marginal distribution function of
and it is denoted by
![[eq2]](/images/marginal_distribution_function__8.png){kind=small}
.
Marginal distribution functions play an important role in the characterization of independence between random variables: two random variables are independent if and only if their joint distribution function is equal to the product of their marginal distribution functions (see the lecture entitled Independent random variables).
Example
Let
and
be two random variables having marginal distribution
functions![[eq3]](/images/marginal_distribution_function__11.png)
and
joint distribution
function![[eq4]](/images/marginal_distribution_function__12.png)
It
is easy to check that
![[eq5]](/images/marginal_distribution_function__13.png){kind=small}
for
any
and
,
which implies that
and
are independent.
A more detailed discussion of the marginal distribution function can be found in the lecture entitled Random vectors.
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Please cite as:
Taboga, Marco (2021). "Marginal distribution function", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/marginal-distribution-function.
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