web.archive.org

Lévy distribution: Information from Answers.com

Lévy (unshifted)

Probability density function
Cumulative distribution function
Parameters	$c > 0\,$
Support	$x \in [0, \infty)$
Probability density function (pdf)	$\sqrt{\frac{c}{2\pi}}~ \frac{e^{-c/2x}}{x^{3/2}}$
Cumulative distribution function (cdf)	$\textrm{erfc}\left(\sqrt{c/2x}\right)$
Mean	infinite
Median	$c/2(\textrm{erf}^{-1}(1/2))^2\,$
Mode	$\frac{c}{3}$
Variance	infinite
Skewness	undefined
Excess kurtosis	undefined
Entropy	$\frac{1+3\gamma+\ln(16\pi c^2)}{2}$
Moment-generating function (mgf)	undefined
Characteristic function	$e^{-\sqrt{-2ict}}$

In probability theory and statistics, the Lévy distribution, named after Paul Pierre Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy this distribution, with frequency as the dependent variable, is known as a Van der Waals profile.

It is one of the few distributions that are stable and that have probability density functions that are analytically expressible, the others being 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.

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

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

where erfc(z) is the complementary error function. A shift parameter μ may be included by replacing each occurrence of x in the above equations with x − μ. This will simply have the effect of shifting the curve to the right by an amount μ, and changing the support to the interval [μ, $\infty$ ). The characteristic function of the Lévy distribution (including a shift μ) is given by

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

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

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

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

$m_n\ \stackrel{\mathrm{def}}{=}\ \sqrt{\frac{c}{2\pi}}\int_0^\infty \frac{e^{-c/2x}\,x^n}{x^{3/2}}\,dx$

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:

$M(t;c)\ \stackrel{\mathrm{def}}{=}\ \sqrt{\frac{c}{2\pi}}\int_0^\infty \frac{e^{-c/2x+tx}}{x^{3/2}}\,dx$

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:

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

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

Probability density function for the Lévy distribution

Related distributions

Applications

The Lévy distribution is of interest to the financial modeling community due to its empirical similarity to the returns of securities.
It is claimed that fruit flies follow a form of the distribution to find food (Lévy flight).^[1]
The frequency of geomagnetic reversals appears to follow a Lévy distribution
The time of hitting a single point (different from the starting point 0) by the Brownian motion has the Lévy distribution.

References and external links

"Information on stable distributions". Retrieved on July 13, 2005. - 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

^ "The Lévy distribution as maximizing one's chances of finding a tasty snack". Retrieved on April 7, 2007.

v • d • e

Probability distributions

Discrete univariate with finite support

Benford · Bernoulli · binomial · categorical · hypergeometric · Rademacher · discrete uniform · Zipf · Zipf-Mandelbrot

Discrete univariate with infinite support

Boltzmann · Conway-Maxwell-Poisson · compound Poisson · discrete phase-type · extended negative binomial · Gauss-Kuzmin · geometric · logarithmic · negative binomial · parabolic fractal · Poisson · Skellam · Yule-Simon · zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Beta · Irwin-Hall · Kumaraswamy · raised cosine · triangular · U-quadratic · uniform · Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,∞)

Beta prime · Burr · chi-square · Coxian · Erlang · exponential · F · Fermi-Dirac · folded normal · Fréchet · Gamma · generalized extreme value · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-square · hyper-exponential · hypoexponential · inverse chi-square (scale inverse chi-square) · inverse Gaussian · inverse gamma · Lévy · log-normal · log-logistic · Maxwell-Boltzmann · Maxwell speed · Nakagami · noncentral chi-square · Pareto · phase-type · Rayleigh · relativistic Breit–Wigner · Rice · Rosin–Rammler · shifted Gompertz · truncated normal · type-2 Gumbel · Weibull · Wilks' lambda

Continuous univariate supported on the whole real line (-∞,∞)

Cauchy · extreme value · exponential power · Fisher's z · Fisher-Tippett · generalized hyperbolic · hyperbolic secant · Landau · Laplace · Lévy skew alpha-stable · logistic · normal (Gaussian) · normal inverse Gaussian · skew normal · Student's t · type-1 Gumbel · Variance-Gamma · Voigt

Multivariate (joint)

Discrete: Ewens · Beta-binomial · multinomial · multivariate Polya Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · multivariate Student · normal-scaled inverse gamma · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart

Directional, degenerate, and singular

Directional: Kent · von Mises · von Mises–Fisher Degenerate: discrete degenerate · Dirac delta function Singular: Cantor

Families

exponential · natural exponential · location-scale · maximum entropy · Pearson · Tweedie

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)