Moments

For any Random variable $X$ and $n=1,2,3,\dots$, we have the following definitions.

• The $n$th moment of $X$ is $\mathbb{E}(X^n)$. When $n=1$, this is just the Expectation.
• The $n$th central moment of $X$ is $\mathbb{E}([X-\mu {]}^n)$, where $\mu =\mathbb{E}(X)$. When $n=2$, this is just the Variance.
• The $n$th standardized moment of $X$ is $\mathbb{E}\left({\left[\frac{X-\mu }{\sigma }\right]}^n\right)$.

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.