## Term Glossary

### Central Moment

Central moment of return distribution is a moment of a probability distribution of returns about expected return: $$\mu_n = E[(r - E[r])^n]$$ For discrete return distributions: $$Var(r)=\sum\limits _{i=1}^{\infty}(r_{i}-E(r))^{n}P(r=r_{i})$$ $$\sum\limits _{i=1}^{\infty}P(r=r_{i})=1$$

$r$
asset return
$E(r)$
expected return
$P(r=r_i)$
probability of i-th return realization
$n$
moment order, non-negative integer
For continous return distributions: $$\mu_n=\intop_{-\infty}^{+\infty}(r-E(r))^{n}f(r)dr$$
$r$
asset return
$E(r)$
expected return
$f(r)$
probability density function of return distribution
$n$
moment order, non-negative integer
The first central moment $\mu_1$ is 0. The second central moment $\mu_2$ is called the variance. The third and fourth central moments are used to define the standardized moments which are used to define skewness and kurtosis, respectively.

###### Function Reference
portfolio_moment, position_moment