This lecture introduces the notion of moment of a random variable.

Table of contents

The -th moment of a random variable is the expected value of its -th power.

Definition
Let
be a random variable. Let
.
If the expected
valueexists
and is finite, then
is said to possess a finite
-th
moment and
is called
the** **-th
**moment**
of** **.
If
is not well-defined, then we say that
does not possess the
-th
moment.

The following example shows how to compute a moment of a discrete random variable.

Example Let be a discrete random variable having supportand probability mass functionThe third moment of can be computed as follows:

The -th central moment of a random variable is the expected value of the -th power of the deviation of from its expected value.

Definition
Let
be a random variable. Let
.
If the expected
valueexists
and is finite, then
is said to possess a finite
-th
central moment and
is called
the** ****-**th
**central moment **of
.

The next example shows how to compute the central moment of a discrete random variable.

Example Let be a discrete random variable having supportand probability mass functionThe expected value of isThe third central moment of can be computed as follows:

The following subsections contain more details about moments.

A generalization of the concept of moment to random vectors is introduced in the lecture entitled Cross-moments.

The moments of a random variable can be easily computed by using either its moment generating function, if it exists, or its characteristic function (see the lectures entitled Moment generating function and Characteristic function).

Please cite as:

Taboga, Marco (2021). "Moments of a random variable", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/fundamentals-of-probability/moments.

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