Moments

(Remember definition of probability density function f(x):

)

Moments​ ​are​ ​measure​ ​of​ ​​probability​ ​density​ (in general - measure of “shape of a set of points”). The​ ​n-th​ ​moment​ ​µ​n​ ​ ​of​ ​a​ ​real-valued​ ​continuous​ ​function​ ​f(x)​ ​about​ ​value​ ​c​ of a probability density function f(x) is defined:

The value c is typically set as the mean of a distribution, then the moments are called “central” moments. If c=0, the moment is called a “raw” moment and is marked as µn‘.

The​ ​zeroth​ ​(raw)​ ​moment​ ​is​ ​equal​ ​to​ ​1​ (total area under the probability density function f(x))

The​ ​first​ ​(raw)​ ​moment​ ​is​ ​the​ ​mean

The​ ​second​ ​(central)​ ​moment​ ​is​ ​the​ ​variance

For the higher moments, standardized variants are typically shown (divided by σ^n ) and are marked as µ ̃.

The​ ​third​ ​(standardized​ ​central)​ ​moment​ ​is​ ​skewness

The​ ​fourth​ ​(standardized​ ​central)​ ​moment​ ​is​ ​kurtosis