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 and a stable distribution.
Definition The probability density function of the Lévy distribution over the domain x ≥ μ {\displaystyle x\geq \mu } is
f ( x ; μ , c ) = c 2 π e − c 2 ( x − μ ) ( x − μ ) 3 / 2 , {\displaystyle f(x;\mu ,c)={\sqrt {\frac {c}{2\pi }}}\,{\frac {e^{-{\frac {c}{2(x-\mu )}}}}{(x-\mu )^{3/2}}},}
where μ {\displaystyle \mu } is the location parameter and c {\displaystyle c} is the scale parameter. The cumulative distribution function is
F ( x ; μ , c ) = erfc ( c 2 ( x − μ ) ) = 2 − 2 Φ ( c ( x − μ ) ) , {\displaystyle F(x;\mu ,c)=\operatorname {erfc} \left({\sqrt {\frac {c}{2(x-\mu )}}}\right)=2-2\Phi \left({\sqrt {\frac {c}{(x-\mu )}}}\right),}
where erfc ( z ) {\displaystyle \operatorname {erfc} (z)} is the complementary error function, and Φ ( x ) {\displaystyle \Phi (x)} is the Laplace function (CDF of the standard normal distribution). The shift parameter μ {\displaystyle \mu } has the effect of shifting the curve to the right by an amount μ {\displaystyle \mu } and changing the support to the interval [ μ {\displaystyle \mu } , ∞ {\displaystyle \infty } ). Like all stable distributions, the Lévy distribution has a standard form f(x; 0, 1) which has the following property:
f ( x ; μ , c ) d x = f ( y ; 0 , 1 ) d y , {\displaystyle f(x;\mu ,c)\,dx=f(y;0,1)\,dy,}
where y is defined as
y = x − μ c . {\displaystyle y={\frac {x-\mu }{c}}.}
The characteristic function of the Lévy distribution is given by
φ ( t ; μ , c ) = e i μ t − − 2 i c t . {\displaystyle \varphi (t;\mu ,c)=e^{i\mu t-{\sqrt {-2ict}}}.}
Note that the characteristic function can also be written in the same form used for the stable distribution with α = 1 / 2 {\displaystyle \alpha =1/2} and β = 1 {\displaystyle \beta =1} :
φ ( t ; μ , c ) = e i μ t − | c t | 1 / 2 ( 1 − i sign ( t ) ) . {\displaystyle \varphi (t;\mu ,c)=e^{i\mu t-|ct|^{1/2}(1-i\operatorname {sign} (t))}.}
Assuming μ = 0 {\displaystyle \mu =0} , the nth moment of the unshifted Lévy distribution is formally defined by
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