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Levy distribution

From phenomenon to model

The family is linked to Paul Lévy’s work on stable laws and random processes. It also arises as the first-passage law of driftless Brownian motion.

What to look at

Its support begins just above mu and continues without bound. c controls tail scale and mu the threshold; classical mean and variance do not exist.

Useful relationships

It is a positive stable-law case and can be obtained from a reciprocal square of a normal variable. It differs from Inverse Gaussian because the first-passage process has no positive drift.

Names you may also encounter

It is also called the Lévy distribution or positive stable distribution with index 1/2.

When it makes sense

It can model extreme durations in diffusion processes or jump sizes. In short samples, a histogram often hides the instability of upper quantiles.

What not to assume

Do not make the sample mean the main summary or compare fits only by their visible tail: high-quantile uncertainty is structural.

A stable law born from a first encounter

The first time driftless Brownian motion reaches a positive level follows a Lévy distribution. The same family is a stable law with index one half and complete right skewness. Adding independent Lévy variables, after suitable rescaling, stays within the family.

Its tail is exceptionally heavy: neither mean nor variance exists. This is not a numerical limitation, and no increase in sample size repairs it within the model. Very large observations belong to the law rather than being automatic anomalies. In transport and first-passage applications the physical story supports that tail; for generic positive data, Lévy can extrapolate far more aggressively than Lognormal, Gamma, or Weibull.

Decision guide

A good candidate when: a first-passage or stable-law mechanism produces a positive variable with an extremely heavy tail and no finite mean.

Compare it with: Inverse Gaussian and Generalized Pareto. If the analysis requires stable averages, Lévy is conceptually incompatible even if it fits some quantiles.

References

Levy Distribution: equations and calculator

Distribution defintion

X∼Levy(μ,c)X\sim\mathrm{Levy}\left(\mu,c\right)

Distribution domain

x∈[μ,∞)x\in [\mu,\infty)

Parameters domain and parameters constraints

μ∈R,c∈R+\mu\in\mathbb{R},c\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=1−erf(c2(x−μ))F_{X}\left(x\right)=1-\textrm{erf}\left(\sqrt{\frac{c}{2(x-\mu)}}\right)

Probability density function

fX(x)=c2π  e−c2(x−μ)(x−μ)3/2f_{X}\left(x\right)=\sqrt{\frac{c}{2\pi}}~~\frac{e^{-\frac{c}{2(x-\mu)}}}{(x-\mu)^{3/2}}

Percent point function/Sample

FX−1(u)=μ+c2(erf−1(1−u))2F^{-1}_{X}\left(u\right)=\mu+\frac{c}{2\left(\textrm{erf}^{-1}(1-u)\right)^2}

Non-central parametric moments

μk′=E[Xk]=∫μ∞xkfX(x)dx\mu'_{k}=E[X^k]=\int_{\mu }^{\infty }x^{k}f_{X}\left(x\right)dx

Parametric mean

Mean(X)=μ1′=∞\mathrm{Mean}(X)=\mu'_{1}=\infty

Parametric variance

Variance(X)=μ2′−μ1′2=∞\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=\infty

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=undefined\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=\text{undefined}

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=undefined\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}=\text{undefined}

Parametric median

Median(X)=μ+c2(erf−1(1/2))2\mathrm{Median}(X)=\mu+\frac{c}{2(\textrm{erf}^{-1}(1/2))^2}

Parametric mode

Mode(X)=μ+c3\mathrm{Mode}(X)=\mu+\frac{c}{3}

Additional information and definitions

μ:Location parameter\mu:\text{Location parameter}
c:Scale parameterc:\text{Scale parameter}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
erf(x):Error function\mathrm{erf}(x):\text{Error function}
erf−1(x):Inverse of error function\mathrm{erf}^{-1}(x):\text{Inverse of error function}