Moments of a random variable

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

Moment

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:

Central moment

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:

More details

The following subsections contain more details about moments.

Multivariate generalization

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

Computation

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).