This distribution is a special case of a stable distribution with shape parameter α = 1/2 and skewness parameter β = 0.
And it does not appear to be widely used.
This name "SαS point5 distribution" is not official but provisional.
Such a distribution with β = 0 is called symmetric alpha-stable distribution.
- α = 2: Normal distribution
- α = 1: Cauchy distribution
The SαS point5 distribution, like these distribution, has a closed-from expression, but it can't be expressed in terms of elementary function.
The SαS point5 distribution, generalized to a stable distribution by introducing position and scale parameters, is as follows:
Since scaling and translations allow for standardization, standard parameters are discussed here.
Using fresnel integral function, it's obtained as follows:
This equation becames difficult to obtain accurately when x is extremely small.
When x is extremely small, the following series expression can also be used:
When x is large, the following equation can be used as an asymptotic expression.
The coefficients decay rapidly, making them suitable numerical evaluation:
| stat | x | note |
|---|---|---|
| mean | N/A | undefined |
| mode | 0 | |
| median | 0 | |
| variance | N/A | undefined |
| 0.75-quantile | 1.283832775189327742808346186911... | |
| 0.9-quantile | 12.74134266157698167806432787517... | |
| 0.95-quantile | 57.30402773063651002282616967046... | |
| 0.99-quantile | 155.9726103725110631926561534085... | |
| entropy | 3.639924445680306495730849603907... |
PDF Precision 150
CDF Precision 150
Quantile Precision 142
