~ \frac{e^{-c/2x}}{x^{3/2}} | cdf =$\backslash textrm\{erfc\}\backslash left(\backslash sqrt\{c/2x\}\backslash right)$| mean =infinite| median =$c/2(\backslash textrm\{erf\}^\{-1\}(1/2))^2\backslash ,$| mode =$\backslash frac\{c\}\{3\}$| variance =infinite| skewness =undefined| kurtosis =undefined| entropy =$\backslash frac\{1+3\backslash gamma+\backslash ln(16\backslash pi\; c^2)\}\{2\}$| mgf =undefined| char =$e^\{-\backslash sqrt\{-2ict\}\}$| }} In probability theory and statistics, the Lévy distribution, named after Paul Pierre Lévy, is one of the few distributions that are stable and that have probability density functions that are analytically expressible. The others are the normal distribution and the Cauchy distribution. All three are special cases of the Lévy skew alpha-stable distribution, which does not generally have an analytically expressible probability density. In spectroscopy this distribution, with frequency as the dependent variable, is known as a Van der Waals profile.

The probability density function of the Lévy distribution over the domain $x\backslash ge\; 0$ is

$$

f(x;c)=\sqrt{\frac{c}{2\pi}}~~\frac{e^{-c/2x}}{x^{3/2}}

where $c$ is the scale parameter. The cumulative distribution function is

$F(x;c)=\backslash textrm\{erfc\}\backslash left(\backslash sqrt\{c/2x\}\backslash right)$

where $\backslash textrm\{erfc\}(z)$ is the complementary error function. A shift parameter $\backslash mu$ may be included by replacing each occurrence of $x$ in the above equations with $x-\backslash mu$. This will simply have the effect of shifting the curve to the right by an amount $\backslash mu$. The characteristic function of the Lévy distribution (including a shift $\backslash mu$) is given by

$\backslash varphi(t;c)=e^\{i\backslash mu\; t-\backslash sqrt\{-2ict\}\}.$

Note that the characteristic function can also be written in the same form used for the Lévy skew alpha-stable distribution with $\backslash alpha=1/2$ and $\backslash beta=1$:

$\backslash varphi(t;c)=e^\{i\backslash mu\; t-|ct|^\{1/2\}~(1-i~\backslash textrm\{sign\}(t))\}.$

The nth moment of the unshifted Lévy distribution is formally defined by:

$m\_n\backslash equiv\backslash sqrt\{\backslash frac\{c\}\{2\backslash pi\}\}\backslash int\_0^\backslash infty\; \backslash frac\{e^\{-c/2x\}\backslash ,x^n\}\{x^\{3/2\}\}\backslash ,dx$

which diverges for all n > 0 so that the moments of the Lévy distribution do not exist. The moment generating function is formally defined by:

$M(t;c)\backslash equiv\; \backslash sqrt\{\backslash frac\{c\}\{2\backslash pi\}\}\backslash int\_0^\backslash infty\; \backslash frac\{e^\{-c/2x+tx\}\}\{x^\{3/2\}\}\backslash ,dx$

which diverges for $t>0$ and is therefore not defined in an interval around zero, so that the moment generating function is not defined per se. In the wings of the distribution, the PDF exhibits heavy tail behavior falling off as:

$\backslash lim\_\{x\backslash rightarrow\; \backslash infty\}f(x;c)\; =\backslash sqrt\{\backslash frac\{c\}\{2\backslash pi\}\}~\backslash frac\{1\}\{x^\{3/2\}\}.$

This is illustrated in the diagram below, in which the PDF's for various values of c are plotted on a log-log scale.

Related distributions

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• Relation to Lévy skew alpha-stable distribution: If $X\; \backslash sim\; \backslash textrm\{Levy\}(c)$ then $X\; \backslash sim\; \backslash textrm\{Levy-S\}\backslash alpha\backslash textrm\{S\}(1/2,1,c,0)$
• Relation to Scale-inverse-chi-square distribution: If $X\; \backslash sim\; \backslash textrm\{Levy\}(c)$ then $X\; \backslash sim\; \backslash textrm\{Scale-inv-\}\backslash chi^2(1,c)$
• Relation to inverse gamma distribution: If $X\; \backslash sim\; \backslash textrm\{Levy\}(c)$ then $X\; \backslash sim\; \backslash textrm\{Inv-Gamma\}(1/2,c/2)$

References and external links

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• - John P. Nolan's introduction to stable distributions, some papers on stable laws, and a free program to compute stable densities, cumulative distribution functions, quantiles, estimate parameters, etc. See especially An introduction to stable distributions, Chapter 1

Continuous distributions

Lévy-Verteilung | Distribution de Lévy