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Holtsmark field distribution

Holtsmark
Probability density function
Symmetric stable distributions
Symmetric α-stable distributions with unit scale factor; α=1.5 (blue line) represents the Holtsmark distribution
Cumulative distribution function
CDF's for symmetric α-stable distributions; α=3/2 represents the Holtsmark distribution
Parameters

c ∈ (0, ∞) — scale parameter

μ ∈ (−∞, ∞) — location parameter
Support xR
PDF expressible in terms of hypergeometric functions; see text
Mean μ
Median μ
Mode μ
Variance infinite
Skewness undefined
Ex. kurtosis undefined
MGF undefined
CF

c ∈ (0, ∞) — scale parameter

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 equal to 3/2 and skewness parameter of zero. Since 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.


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Wikipedia

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