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

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).

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