This lecture defines the notion of cross-moment of a random vector, which is a generalization of the concept of moment of a random variable (see the lecture entitled Moments of a random variable).
Table of contents
Let
be a
random vector. A cross-moment of
is the expected value of the product of integer
powers of the entries of
:where
is the
-th
entry of
and
are non-negative integers.
The following is a formal definition of cross-moment.
Definition
Let
be a
random vector. Let
and
.
If![[eq5]](/images/cross-moments__14.png){kind=small}
exists
and is finite, then it is called a cross-moment of
of order
.
If all cross-moments of order
exist and are finite, i.e. if (1) exists and is finite for all
-tuples
of non-negative integers
such that
,
then
is said to possess finite cross-moments of order
.
The following example shows how to compute a cross-moment of a discrete random vector.
Example
Let
be a
discrete random vector and denote its components by
,
and
.
Let the support of
be
![[eq8]](/images/cross-moments__29.png)
and
its joint probability
mass function
be![[eq9]](/images/cross-moments__30.png)
The
following is a cross-moment of
of order
:which
can be computed by using the
transformation
theorem:![[eq11]](/images/cross-moments__34.png)
The central cross-moments of a random vector
are just the cross-moments of the random vector of deviations
.
Definition
Let
be a
random vector. Let
and
.
If![[eq15]](/images/cross-moments__41.png){kind=small}
exists
and is finite, then it is called a central cross-moment of
of order
.
If all central cross-moments of order
exist and are finite, that is, if (2) exists and is finite for all
-tuples
of non-negative integers
such that
,
then
is said to possess finite central cross-moments of order
.
The following example shows how to compute a central cross-moment of a discrete random vector.
Example
Let
be a
discrete random vector and denote its components by
,
and
.
Let the support of
be![[eq18]](/images/cross-moments__56.png)
and
its joint probability mass function
be![[eq19]](/images/cross-moments__57.png)
The
expected values of the three components of
areThe
following is a central cross-moment of
of order
:![[eq21]](/images/cross-moments__62.png){kind=small}
which
can be computed by using the transformation
theorem:![[eq22]](/images/cross-moments__63.png)
Please cite as:
Taboga, Marco (2021). "Cross-moments of a random vector", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/fundamentals-of-probability/cross-moments.
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