Moment

$m_n^p:=\frac{1}{n}{\sum }_{i=1}^n{x}_i^p$

• $p=1$ Arithmetic Mean
• $p=-1$ harmonic mean

The Expectation $E_F(X^p)$ is then called the $p^{th}$ moment of $F$.

We have

$$\frac{1}{n}\sum _{i=1}^n{x}_i^r⟶{\mathbb{E}}_F\left(X^r\right)$$

hence the moment can be used to estimate the Expectation of an ECDF.

We also get a 95% Confidence Interval

$$\left|m_n^p-{\mathbb{E}}_F\left(X^p\right)\right|\le \frac{2\sqrt{{\mathbb{V}}_F\left(X^p\right)}}{\sqrt{n}}$$