and 
are termed cumulative distribution
functions and are feasible for ion implantation profiles. Furthermore,
F(x) must be a continuous function, so that we can write the distribution
function as
where f(t) is the probability density function. Combining the relation
and (2.1-1) the final concentration C(x) of an implanted
dopant is given by
where 
is the total implantation dose per unit area. It depends on
the physical properties of the implanted ions what kind of probability
density function is chosen for which application. In the case of channeling
implantation profiles probability density functions based on central moments
totally fail, but they give acceptable results for amorphous implantations.
Most of the simulators used for ion implantation are limited to conventional
central moments and their characteristic parameters, but there are still
other statistical moments available and seem to be worth for further
investigations. In the following sections we will discuss the central
moments and the probability weighted moments as vehicles to reproduce ion
implantation profile by statistical means.
