In probability theory and statistics, the Lévy distribution, named after Paul 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 a special case of the inverse-gamma distribution. It is a stable distribution.

The probability density function of the Lévy distribution over the domain ${\displaystyle x\geq \mu }$ is

where ${\displaystyle \mu }$ is the location parameter, and ${\displaystyle c}$ is the scale parameter. The cumulative distribution function is

where ${\displaystyle \operatorname {erfc} (z)}$ is the complementary error function, and ${\displaystyle \Phi (x)}$ is the Laplace function (CDF of the standard normal distribution). The shift parameter ${\displaystyle \mu }$ has the effect of shifting the curve to the right by an amount ${\displaystyle \mu }$ and changing the support to the interval [${\displaystyle \mu }$, ${\displaystyle \infty }$). Like all stable distributions, the Lévy 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 ${\displaystyle \alpha =1/2}$ and ${\displaystyle \beta =1}$:

Assuming ${\displaystyle \mu =0}$, the nth moment of the unshifted Lévy distribution is formally defined by