A discrete random vector is a random vector that can take either a finite or an infinite but countable number of values.
Table of contents
The following is a possible definition.
Definition
A
random vector
is said to be discrete if and only if the set of values it can take has either
a finite or an infinite but countable number of elements, and there exists a
function
![[eq1]](/images/discrete_random_vector__3.png){kind=small}
,
called joint probability mass function, such
that![[eq2]](/images/discrete_random_vector__4.png){kind=small}
where
![[eq3]](/images/discrete_random_vector__5.png){kind=small}
is the probability that
takes the value
.
A random vector that is discrete is also said to possess a multivariate discrete distribution.
Suppose a
random vector
can take only one of three values,
,
or
defined
by![[eq4]](/images/discrete_random_vector__13.png)
and
that
has probability
of being observed, while
and
have probability
.
Then,
is discrete because it can take only finitely many values. Its joint
probability mass function
is![[eq5]](/images/discrete_random_vector__20.png)
When
the two entries of
are denoted by
and
,
the joint probability mass function can also be written
as![[eq6]](/images/discrete_random_vector__24.png)
The lecture entitled Random vectors provides a more complete treatment of discrete random vectors and joint probability mass functions.
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Please cite as:
Taboga, Marco (2021). "Discrete random vector", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/discrete-random-vector.
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