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Moments of a random variable

by , PhD

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).

How to cite

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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