Lévy Distribution

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."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.

Definition

The probability density function of the Lévy distribution over the domain $x\ge \mu$ is

:$f(x;\mu,c)=\sqrt~~\frac$

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

:$F(x;\mu,c)=1 - \textrm\left(\sqrt\right)= 2-2\Phi \left(\right)$

where $\textrm(z)$ is the complementary error function and $\Phi (x)$ is the Laplace Function (CDF of the Standard Normal Distribution). 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_distributions,_the_Levy_distribution_has_a_standard_form_f(x;0,1)_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_(probability_theory).html" "title="math>\mu, $\infty$). Like all stable distributions, the Levy distribution has a standard form f(x;0,1) 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).html" "title="characteristic function (probability theory)">characteristic function 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