where is the complementary
error function, and is the Laplace function (
CDF of the standard
normal distribution). The shift parameter has the effect of shifting the curve to the right by an amount and changing the support to the interval [, ). Like all
stable distributions, the Lévy distribution has a standard form f(x; 0, 1) which has the following property:
however, this diverges for and is therefore not defined on an interval around zero, so the moment-generating function is actually undefined.
Like all
stable distributions except the
normal distribution, the wing of the probability density function exhibits heavy tail behavior falling off according to a power law:
as
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 are plotted on a
log–log plot:
The standard Lévy distribution satisfies the condition of being
stable:
where are independent standard Lévy-variables with
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[1]
is Lévy-distributed with location and scale . Here is the cumulative distribution function of the standard
normal distribution.
The
time of hitting a single point, at distance from the starting point, by the
Brownian motion has the Lévy distribution with . (For a Brownian motion with drift, this time may follow an
inverse Gaussian distribution, 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.[2]
^"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,
ISBN0-471-27663-4,
ISBN978-0-471-27663-0,
[1]; and in Journal of technical physics, Volume 36, by Instytut Podstawowych Problemów Techniki (Polska Akademia Nauk), publisher: Państwowe Wydawn. Naukowe., 1995,
[2]
Notes
^"The Lévy Distribution". Random. Probability, Mathematical Statistics, Stochastic Processes. The University of Alabama in Huntsville, Department of Mathematical Sciences. Archived from
the original on 2017-08-02.