= Moment =

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

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== 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:

{{attachment:first.svg}}

Compare to the [[Calculus/AverageFunctionValue|average function value]] of the same function ''f'':

{{attachment:ave.svg}}

These are similar but subtly different calculations.

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== Random Variables ==

Moments are used to describe distributions of random variables.



=== First Moment ===

For a probability distribution, the first moment is the [[Analysis/ExpectedValue|expected value]].

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

{{attachment:cont1.svg}}

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:

{{attachment:dis1.svg}}

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:

{{attachment:cont2.svg}}

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.

{{attachment:cont2center.svg}}

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 [[Analysis/Variance|variance]].



=== Nth Moment ===

Abstractly, the ''n''th moment of a continuous quantitative random variable is:

{{attachment:contn.svg}}

And the ''n''th moment of a discrete quantitative random variable is:

{{attachment:disn.svg}}

When the ''n''th central moment is divided by the ''n''th power of [[Analysis/Variance|standard deviation]], it is known as the ''n''th '''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'''.



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