Editing Stable distribution (section)

=== Parametrizations ===
The parametrization of stable distributions is not unique. Nolan <ref name="Nolan2020" /> tabulates 11 parametrizations seen in the literature and gives conversion formulas. The two most commonly used parametrizations are the one above (Nolan's "1") and the one immediately below (Nolan's "0").

The parametrization above is easiest to use for theoretical work, but its probability density is not continuous in the parameters at <math>\alpha =1</math>.<ref name="Nolan 1997"/> A continuous parametrization, better for numerical work,  is<ref name=":0" />
<math display="block"> \varphi(t; \alpha, \beta, \gamma, \delta) = \exp \left (i t \delta - |\gamma t|^\alpha \left (1 - i \beta \sgn(t) \Phi \right ) \right ) </math>
where:
<math display="block"> \Phi = \begin{cases} \left ( |\gamma t|^{1 - \alpha} - 1 \right ) \tan \left (\tfrac{\pi \alpha}{2} \right )  & \alpha \neq 1 \\ - \frac{2}{\pi} \log|\gamma t| & \alpha = 1 \end{cases} </math>

The ranges of <math>\alpha</math> and <math>\beta</math> are the same as before, ''γ'' (like ''c'') should be positive, and ''δ'' (like ''μ'') should be real.

In either parametrization one can make a linear transformation of the random variable to get a random variable whose density is <math> f(y; \alpha, \beta, 1, 0) </math>. In the first parametrization, this is done by defining the new variable:
<math display="block"> y = \begin{cases} \frac{x - \mu}\gamma & \alpha \neq 1 \\  \frac{x - \mu}\gamma - \beta\frac 2\pi\ln\gamma  & \alpha = 1 \end{cases} </math>

For the second parametrization, simply use
<math display="block"> y = \frac{x-\delta}\gamma </math>
independent of <math>\alpha</math>. In the first parametrization, if the mean exists (that is, <math>\alpha > 1</math>) then it is equal to ''μ'', whereas in the second parametrization when the mean exists it is equal to <math> \delta - \beta \gamma \tan \left (\tfrac{\pi\alpha}{2} \right).</math>