Raw moments versus central moments

The raw moments (or 'moments about zero') of a distribution are defined as

for continuous distributions with PDF f(x) and

for discrete distributions with PMF pi.

The central moments (or 'moments about the mean') for are defined as:

with analogue definitions for discrete variables. The lower central moments are directly related to the variance, skewness and kurtosis.

The second, third and fourth central moments can be expressed in terms of the raw moments as follows:

ModelRisk allows one to directly calculate all four raw moments of a distribution object through the VoseRawMoments function.