The i-th non-central moments of a probability density function f(x) are defined by:
By introducing a bias, practically the mean value of the distribution, we get the central moments:
The characteristic parameters such as projected range 
, standard
deviation 
, skewness 
, and kurtosis 
can be
expressed in terms of these central moments:
It is also possible to define the characteristic parameters ,
, , and directly by their probability density
function f(x), as can be found in [Wim93]. Several density
distribution functions are based on central moments, like the most popular
Gaussian distribution (2.1-9 ). It uses only two characteristic
parameters and , and the approximation of ion implantation
profiles can therefore be rather inaccurate.
Especially for nonsymmetric ion implantation profiles the lack of accuracy in the tail of the density functions constrain the usage of other functions like the Joined Half Gaussian density function [Gib73] or the Pearson density function [Hof75a].
The Joined Half Gaussian density function is defined by two Gaussian functions, which join at a modal projected range. The major drawback of this density function is a restricted range of skewness values. Better fits can be achieved by the Pearson density function family. The Pearson density functions are derived as solutions of the differential equation (2.1-10).
The Pearson coefficients a, 
, 
, and 
can be expressed in terms
of the first four characteristic parameters:
Depending on the values of and , seven different solutions
for the Pearson family can be obtained. For practical implantation
applications the Pearson IV density function is frequently used [Rys81]
[Sel84]. By combining two Pearson density functions, using one for the
surface region and the other one for the bulk, it is also possible to model
typical channeling tails of implantation profiles [Par90] [Yan95].
