Holtsmark distribution

Holtsmark

Probability density function Symmetric α-stable distributions with unit scale factor; α=1.5 (blue line) represents the Holtsmark distribution
Cumulative distribution function
Parameters	c ∈ (0, ∞) — scale parameter μ ∈ (−∞, ∞) — location parameter
Support	x ∈ R
PDF	expressible in terms of hypergeometric functions; see text
Mean	μ
Median	μ
Mode	μ
Variance	infinite
Skewness	undefined
Excess kurtosis	undefined
MGF	undefined
CF	${\displaystyle \exp \left[~it\mu \!-\!|ct|^{3/2}~\right]}$

The (one-dimensional) Holtsmark distribution is a continuous probability distribution. The Holtsmark distribution is a special case of a stable distribution with the index of stability or shape parameter ${\displaystyle \alpha }$ equal to 3/2 and the skewness parameter ${\displaystyle \beta }$ of zero. Since ${\displaystyle \beta }$ equals zero, the distribution is symmetric, and thus an example of a symmetric alpha-stable distribution. The Holtsmark distribution is one of the few examples of a stable distribution for which a closed form expression of the probability density function is known. However, its probability density function is not expressible in terms of elementary functions; rather, the probability density function is expressed in terms of hypergeometric functions.

The Holtsmark distribution has applications in plasma physics and astrophysics.^[1] In 1919, Norwegian physicist Johan Peter Holtsmark proposed the distribution as a model for the fluctuating fields in plasma due to the motion of charged particles.^[2] It is also applicable to other types of Coulomb forces, in particular to modeling of gravitating bodies, and thus is important in astrophysics.^[3][4]