A momentary diversion

(Aug 31, 2016)

An important quantity in mathematical statistics is the moment of a distribution, i.e. the expected value of a given power of the observations.  Moments can be either raw, centred about a particular value or standardised in some way.  The simplest example is the mean of a distribution: this is the raw first moment, i.e. the expected value of each observation raised to the power 1:

$\mu = {\rm E}[X] = \int_{-\infty}^{\infty}xf(x)dx$

The next example is the variance of a distribution: this is the second central moment, i.e. the expected value of the second power of the difference between each observation and the mean:

$\sigma^2 = {\rm Var}[X] = \int_{-\infty}^{\infty}(x-\mu)^2f(x)dx$

Other central moments…