The graphic shows the value of the cumulative probability distribution function at
of a member of a family of standardized stable distributions parametrized by
. There are no explicit formulas for the general case, so calculations are made using numerical integration of the inverse Fourier transform of the characteristic function. The usual representation of this integrand is highly oscillatory. The Zolotarev transform of the integrand is a slowly varying function over a fixed interval. In this representation, total probability is represented as the area of a unit square. The green area is the cumulative probability, the orange area is 1 minus this probability. The integral works only for
, but values for
can be obtained by the symmetrical properties of stable distributions; at this point you will see the colors in the graph reverse. The function has a closed form at
, so this point can always be calculated when
, the integral formula is different, but does not exist when
is the shape parameter; when
, the distribution is a Normal distribution with
has a range of
; in the Demonstration it is supported down to 0.1.
This Demonstration was created as a first step in developing faster numerical methods for calculating stable distributions, perhaps by methods similar to those in plotting graphics. The idea of the graphic is to view the area as a Lebesgue integral: a set of points with each point reflecting a measure of area.