Moment

A moment is a description of a function's shape.

Description

Functions can be described by singular measures that relate to their shape. For example, the first moment of a locally integrable function f that is defined between a and b is:

Compare to the average function value of the same function f:

These are similar but subtly different calculations.

Random Variables

Moments are used to describe distributions of random variables.

First Moment

For a probability distribution, the first moment is the expected value.

The first moment of a continuous quantitative (numeric) random variable is expressed and notated as:

where Ω is the sample space and f is the probability density function for that distribution.

The first moment of a discrete quantitative random variable is:

where Ω is the set of possible values and f is the probability mass function for that distribution (although the distributions of discrete random variables are commonly given as tables, rather than mathematical functions).

Second Moment

The second (raw) moment of a continuous quantitative random variable is:

This measure isn't usually used directly however. More useful is the second centered moment, which is to say the second moment around the center.

The most straightforward center for a distribution is the first moment itself, and by substituting that in for c above you arrive at the formal definition of variance.

Nth Moment

Abstractly, the nth moment of a continuous quantitative random variable is:

And the nth moment of a discrete quantitative random variable is:

When the nth central moment is divided by the nth power of standard deviation, it is known as the nth standardized moment. The effect of this 'standardization' is to create a unit-less measure. The standardized third moment is skewness. The standardized fourth moment is kurtosis.

CategoryRicottone