In probability theory, the Holtsmark distribution is a probability distribution named after Johan Peter Holtsmark, In 1919.
It's especially used in astrophysics for modeling gravitational bodies.
This distribution is a special case of a stable distribution with shape parameter α = 3/2 and skewness parameter β = 0.
Such a distribution with β = 0 is called symmetric alpha-stable distribution.
- α = 2: Normal distribution
- α = 1: Cauchy distribution
The Holtsmark distribution, like these distribution, has a closed-from expression, but it can't be expressed in terms of elementary function.
The Holtsmark 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 hypergeometric function, it's obtained as follows:
The following series expression can also be used:
When x is large, the following equation can be used as an asymptotic expression:
Evaluating only one term of this equation yields the tail power of the distribution:
| stat | x | note |
|---|---|---|
| mean | 0 | |
| mode | 0 | |
| median | 0 | |
| variance | N/A | undefined |
| 0.75-quantile | 0.9689331817135830052087863130496... | |
| 0.9-quantile | 2.061462638139193764464365388228... | |
| 0.95-quantile | 3.051940973238316697189330345356... | |
| 0.99-quantile | 7.736446206485418555263326182069... | |
| entropy | 2.069448505134624400315580038454... |
PDF Precision 150
CDF Precision 150
Quantile Precision 142
J.Holtsmark, "Uber die Verbreiterung von Spektrallinien" (1919)
J.C.Pain "Expression of the Holtsmark function in terms of hypergeometric 2F2 and Airy Bi functions" (2010)