For any Random variable and , we have the following definitions.

  • The th moment of is . When , this is just the Expectation.
  • The th central moment of is , where . When , this is just the Variance.
  • The th standardized moment of is .

Moments are often used to summarize properties about a distribution. If expectations are undefined, then so are the moments.

  • The expectation and variance are the 1st moment and 2nd central moments, respectively.
  • Skewness is the 3rd standardized moment, and excess kurtosis is the 4th standardized moment, minus 3.

Related to moments are Moment generating functions.