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Index > Fundamentals of probability

Moments of a random variable

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

Table of Contents

Moment

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

Definition Let X be a random variable. Let $nin U{2115} $. If the expected value[eq1]exists and is finite, then X is said to possess a finite n-th moment and [eq2] is called the n-th moment of X. If [eq3] is not well-defined, then we say that X does not possess the n-th moment.

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

Example Let X be a discrete random variable having support[eq4]and probability mass function[eq5]The third moment of X can be computed as follows:[eq6]

Central moment

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

Definition Let X be a random variable. Let $nin U{2115} $. If the expected value[eq7]exists and is finite, then X is said to possess a finite n-th central moment and [eq8] is called the n-th central moment of X.

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

Example Let X be a discrete random variable having support[eq9]and probability mass function[eq10]The expected value of X is[eq11]The third central moment of X can be computed as follows:[eq12]

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).

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