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<title>The Levy skew alpha-Stable Distribution - GNU Scientific Library -- Reference Manual</title>
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Next: <a rel="next" accesskey="n" href="The-Gamma-Distribution.html">The Gamma Distribution</a>,
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<h3 class="section">19.13 The Levy skew alpha-Stable Distribution</h3>
<div class="defun">
— Function: double <b>gsl_ran_levy_skew</b> (<var>const gsl_rng * r, double c, double alpha, double beta</var>)<var><a name="index-gsl_005fran_005flevy_005fskew-1646"></a></var><br>
<blockquote><p><a name="index-Levy-distribution_002c-skew-1647"></a><a name="index-Skew-Levy-distribution-1648"></a>This function returns a random variate from the Levy skew stable
distribution with scale <var>c</var>, exponent <var>alpha</var> and skewness
parameter <var>beta</var>. The skewness parameter must lie in the range
[-1,1]. The Levy skew stable probability distribution is defined
by a fourier transform,
<pre class="example"> p(x) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^alpha (1-i beta sign(t) tan(pi alpha/2)))
</pre>
<p class="noindent">When \alpha = 1 the term \tan(\pi \alpha/2) is replaced by
-(2/\pi)\log|t|. There is no explicit solution for the form of
p(x) and the library does not define a corresponding <code>pdf</code>
function. For \alpha = 2 the distribution reduces to a Gaussian
distribution with <!-- {$\sigma = \sqrt{2} c$} -->
\sigma = \sqrt{2} c and the skewness parameter has no effect.
For \alpha < 1 the tails of the distribution become extremely
wide. The symmetric distribution corresponds to \beta =
0.
<p>The algorithm only works for <!-- {$0 < \alpha \le 2$} -->
0 < alpha <= 2.
</p></blockquote></div>
<p>The Levy alpha-stable distributions have the property that if N
alpha-stable variates are drawn from the distribution p(c, \alpha,
\beta) then the sum Y = X_1 + X_2 + \dots + X_N will also be
distributed as an alpha-stable variate,
<!-- {$p(N^{1/\alpha} c, \alpha, \beta)$} -->
p(N^(1/\alpha) c, \alpha, \beta).
<!-- PDF not available because there is no analytic expression for it -->
<!-- @deftypefun double gsl_ran_levy_pdf (double @var{x}, double @var{mu}) -->
<!-- This function computes the probability density @math{p(x)} at @var{x} -->
<!-- for a symmetric Levy distribution with scale parameter @var{mu} and -->
<!-- exponent @var{a}, using the formula given above. -->
<!-- @end deftypefun -->
<pre class="sp">
</pre>
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