HOME

TheInfoList



Click Here for Items Related To - Lévy distribution
OR:
Lévy distribution on:   [Wikipedia]   [Google]   [Amazon]

In
probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
and
statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
, the Lévy distribution, named after Paul Lévy, is a
continuous probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
for a non-negative
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
. In
spectroscopy Spectroscopy is the field of study that measures and interprets electromagnetic spectra. In narrower contexts, spectroscopy is the precise study of color as generalized from visible light to all bands of the electromagnetic spectrum. Spectro ...
, this distribution, with frequency as the dependent variable, is known as a van der Waals profile."van der Waals profile" appears with lowercase "van" in almost all sources, such as: ''Statistical mechanics of the liquid surface'' by Clive Anthony Croxton, 1980, A Wiley-Interscience publication, ,

and in ''Journal of technical physics'', Volume 36, by Instytut Podstawowych Problemów Techniki (Polska Akademia Nauk), publisher: Państwowe Wydawn. Naukowe., 1995

/ref> It is a special case of the inverse-gamma distribution. It is a
stable distribution In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be st ...
.


Definition

The
probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
of the Lévy distribution over the domain x \ge \mu is : f(x; \mu, c) = \sqrt \, \frac, where \mu is the
location parameter In statistics, a location parameter of a probability distribution is a scalar- or vector-valued parameter x_0, which determines the "location" or shift of the distribution. In the literature of location parameter estimation, the probability distr ...
, and c is the scale parameter. The cumulative distribution function is : F(x; \mu, c) = \operatorname\left(\sqrt\right) = 2 - 2 \Phi\left(\right), where \operatorname(z) is the complementary
error function In mathematics, the error function (also called the Gauss error function), often denoted by , is a function \mathrm: \mathbb \to \mathbb defined as: \operatorname z = \frac\int_0^z e^\,\mathrm dt. The integral here is a complex Contour integrat ...
, and \Phi(x) is the Laplace function ( CDF of the standard
normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...
). The shift parameter \mu has the effect of shifting the curve to the right by an amount \mu and changing the support to the interval
stable distribution In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be st ...
s, the Lévy distribution has a standard form which has the following property: : f(x; \mu, c) \,dx = f(y; 0, 1) \,dy, where ''y'' is defined as : y = \frac. The characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the function \mathbf_A\colon X \to \, which for a given subset ''A'' of ''X'', has value 1 at points ...
of the Lévy distribution is given by : \varphi(t; \mu, c) = e^. Note that the characteristic function can also be written in the same form used for the stable distribution with \alpha = 1/2 and \beta = 1: : \varphi(t; \mu, c) = e^. Assuming \mu = 0, the ''n''th moment (mathematics)">moment of the unshifted Lévy distribution is formally defined by : m_n\ \stackrel\ \sqrt \int_0^\infty \frac \,dx, which diverges for all n \geq 1/2, so that the integer moments of the Lévy distribution do not exist (only some fractional moments). The moment-generating function would be formally defined by : M(t; c)\ \stackrel\ \sqrt \int_0^\infty \frac \,dx, however, this diverges for t > 0 and is therefore not defined on an interval around zero, so the moment-generating function is actually undefined. Like all
stable distribution In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be st ...
s except the
normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...
, the wing of the probability density function exhibits heavy tail behavior falling off according to a power law: : f(x; \mu, c) \sim \sqrt \, \frac as x \to \infty, which shows that the Lévy distribution is not just heavy-tailed but also fat-tailed. This is illustrated in the diagram below, in which the probability density functions for various values of ''c'' and \mu = 0 are plotted on a
log–log plot In science and engineering, a log–log graph or log–log plot is a two-dimensional graph of numerical data that uses logarithmic scales on both the horizontal and vertical axes. Exponentiation#Power_functions, Power functions – relationshi ...
: : The standard Lévy distribution satisfies the condition of being
stable A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed. Styles There are many different types of stables in use tod ...
: : (X_1 + X_2 + \dotsb + X_n) \sim n^X, where X_1, X_2, \ldots, X_n, X are independent standard Lévy-variables with \alpha = 1/2.


Related distributions

* If X \sim \operatorname(\mu, c), then kX + b \sim \operatorname(k\mu + b, kc). * If X \sim \operatorname(0, c), then X \sim \operatorname(1/2, c/2) ( inverse gamma distribution). Here, the Lévy distribution is a special case of a Pearson type V distribution. * If Y \sim \operatorname(\mu, \sigma^2) (
normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...
), then (Y - \mu)^ \sim \operatorname(0, 1/\sigma^2). * If X \sim \operatorname(\mu, 1/\sqrt), then (X - \mu)^ \sim \operatorname(0, \sigma). * If X \sim \operatorname(\mu, c), then X \sim \operatorname(1/2, 1, c, \mu) (
stable distribution In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be st ...
). * If X \sim \operatorname(0, c), then X\,\sim\,\operatorname(1, c) ( scaled-inverse-chi-squared distribution). * If X \sim \operatorname(\mu, c), then (X - \mu)^ \sim \operatorname(0, 1/\sqrt) ( folded normal distribution).


Random-sample generation

Random samples from the Lévy distribution can be generated using inverse transform sampling. Given a random variate ''U'' drawn from the uniform distribution on the unit interval (0, 1], the variate ''X'' given by : X = F^(U) = \frac + \mu is Lévy-distributed with location \mu and scale c. Here \Phi(x) is the cumulative distribution function of the standard
normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...
.


Applications

* The frequency of
geomagnetic reversal A geomagnetic reversal is a change in the Earth's Dipole magnet, dipole magnetic field such that the positions of magnetic north and magnetic south are interchanged (not to be confused with North Pole, geographic north and South Pole, geograp ...
s appears to follow a Lévy distribution *The time of hitting a single point, at distance \alpha from the starting point, by the
Brownian motion Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
has the Lévy distribution with c=\alpha^2. (For a Brownian motion with drift, this time may follow an
inverse Gaussian distribution In probability theory, the inverse Gaussian distribution (also known as the Wald distribution) is a two-parameter family of continuous probability distributions with support (mathematics), support on (0,∞). Its probability density function is ...
, which has the Lévy distribution as a limit.) * The length of the path followed by a photon in a turbid medium follows the Lévy distribution. * A Cauchy process can be defined as a
Brownian motion Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
subordinated to a process associated with a Lévy distribution.


Footnotes


Notes


References

* - 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 especiall
An introduction to stable distributions, Chapter 1


External links

* {{DEFAULTSORT:Levy distribution Continuous distributions Probability distributions with non-finite variance Power laws Stable distributions Paul Lévy (mathematician)