In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a nonnegative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is known as a van der Waals profile.^{[note 1]} It is a special case of the inversegamma distribution.
It is one of the few distributions that are stable and that have probability density functions that can be expressed analytically, the others being the normal distribution and the Cauchy distribution.
Lévy (unshifted)  

Probability density function


Cumulative distribution function


Parameters  location; scale 
Support  
CDF  
Mean  
Median  , for 
Mode  , for 
Variance  
Skewness  undefined 
Ex. kurtosis  undefined 
Entropy 
where is Euler's constant 
MGF  undefined 
CF 
The probability density function of the Lévy distribution over the domain is
where is the location parameter and is the scale parameter. The cumulative distribution function is
where is the complementary error function. 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 Levy distribution has a standard form f(x;0,1) which has the following property:
where y is defined as
The characteristic function of the Lévy distribution is given by
Note that the characteristic function can also be written in the same form used for the stable distribution with and :
Assuming , the nth moment of the unshifted Lévy distribution is formally defined by:
which diverges for all n > 0 so that the moments of the Lévy distribution do not exist. The moment generating function is then formally defined by:
which diverges for and is therefore not defined in an interval around zero, so that the moment generating function is not defined per se. 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:
(This shows that Lévy is not just Heavytailed but also Fattailed.)
This is illustrated in the diagram below, in which the probability density functions for various values of c and are plotted on a loglog scale.
The standard Lévy distribution satisfies the condition
where are independent standard Lévyvariables 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 Lévydistributed with location and scale . Here is the cumulative distribution function of the standard normal distribution.
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