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<title>GNU Scientific Library – Reference Manual: The Levy skew alpha-Stable Distribution</title>
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<a name="The-Levy-skew-alpha_002dStable-Distribution"></a>
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<p>
Next: <a href="The-Gamma-Distribution.html#The-Gamma-Distribution" accesskey="n" rel="next">The Gamma Distribution</a>, Previous: <a href="The-Levy-alpha_002dStable-Distributions.html#The-Levy-alpha_002dStable-Distributions" accesskey="p" rel="previous">The Levy alpha-Stable Distributions</a>, Up: <a href="Random-Number-Distributions.html#Random-Number-Distributions" accesskey="u" rel="up">Random Number Distributions</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<hr>
<a name="The-Levy-skew-alpha_002dStable-Distribution-1"></a>
<h3 class="section">20.14 The Levy skew alpha-Stable Distribution</h3>
<dl>
<dt><a name="index-gsl_005fran_005flevy_005fskew"></a>Function: <em>double</em> <strong>gsl_ran_levy_skew</strong> <em>(const gsl_rng * <var>r</var>, double <var>c</var>, double <var>alpha</var>, double <var>beta</var>)</em></dt>
<dd><a name="index-Levy-distribution_002c-skew"></a>
<a name="index-Skew-Levy-distribution"></a>
<p>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
<em>[-1,1]</em>. The Levy skew stable probability distribution is defined
by a Fourier transform,
</p>
<div class="example">
<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></div>
<p>When <em>\alpha = 1</em> the term <em>\tan(\pi \alpha/2)</em> is replaced by
<em>-(2/\pi)\log|t|</em>. There is no explicit solution for the form of
<em>p(x)</em> and the library does not define a corresponding <code>pdf</code>
function. For <em>\alpha = 2</em> the distribution reduces to a Gaussian
distribution with <em>\sigma = \sqrt{2} c</em> and the skewness parameter has no effect.
For <em>\alpha < 1</em> the tails of the distribution become extremely
wide. The symmetric distribution corresponds to <em>\beta =
0</em>.
</p>
<p>The algorithm only works for <em>0 < alpha <= 2</em>.
</p></dd></dl>
<p>The Levy alpha-stable distributions have the property that if <em>N</em>
alpha-stable variates are drawn from the distribution <em>p(c, \alpha,
\beta)</em> then the sum <em>Y = X_1 + X_2 + \dots + X_N</em> will also be
distributed as an alpha-stable variate,
<em>p(N^(1/\alpha) c, \alpha, \beta)</em>.
</p>
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