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<title>GNU Scientific Library &ndash; Reference Manual: The Gaussian Tail Distribution</title>

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<a name="The-Gaussian-Tail-Distribution"></a>
<div class="header">
<p>
Next: <a href="The-Bivariate-Gaussian-Distribution.html#The-Bivariate-Gaussian-Distribution" accesskey="n" rel="next">The Bivariate Gaussian Distribution</a>, Previous: <a href="The-Gaussian-Distribution.html#The-Gaussian-Distribution" accesskey="p" rel="previous">The Gaussian Distribution</a>, Up: <a href="Random-Number-Distributions.html#Random-Number-Distributions" accesskey="u" rel="up">Random Number Distributions</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="The-Gaussian-Tail-Distribution-1"></a>
<h3 class="section">20.3 The Gaussian Tail Distribution</h3>
<dl>
<dt><a name="index-gsl_005fran_005fgaussian_005ftail"></a>Function: <em>double</em> <strong>gsl_ran_gaussian_tail</strong> <em>(const gsl_rng * <var>r</var>, double <var>a</var>, double <var>sigma</var>)</em></dt>
<dd><a name="index-Gaussian-Tail-distribution"></a>
<p>This function provides random variates from the upper tail of a Gaussian
distribution with standard deviation <var>sigma</var>.  The values returned
are larger than the lower limit <var>a</var>, which must be positive.  The
method is based on Marsaglia&rsquo;s famous rectangle-wedge-tail algorithm (Ann. 
Math. Stat. 32, 894&ndash;899 (1961)), with this aspect explained in Knuth, v2,
3rd ed, p139,586 (exercise 11).
</p>
<p>The probability distribution for Gaussian tail random variates is,
</p>
<div class="example">
<pre class="example">p(x) dx = {1 \over N(a;\sigma) \sqrt{2 \pi \sigma^2}} \exp (- x^2/(2 \sigma^2)) dx
</pre></div>

<p>for <em>x &gt; a</em> where <em>N(a;\sigma)</em> is the normalization constant,
</p>
<div class="example">
<pre class="example">N(a;\sigma) = (1/2) erfc(a / sqrt(2 sigma^2)).
</pre></div>

</dd></dl>

<dl>
<dt><a name="index-gsl_005fran_005fgaussian_005ftail_005fpdf"></a>Function: <em>double</em> <strong>gsl_ran_gaussian_tail_pdf</strong> <em>(double <var>x</var>, double <var>a</var>, double <var>sigma</var>)</em></dt>
<dd><p>This function computes the probability density <em>p(x)</em> at <var>x</var>
for a Gaussian tail distribution with standard deviation <var>sigma</var> and
lower limit <var>a</var>, using the formula given above.
</p></dd></dl>

<br>

<dl>
<dt><a name="index-gsl_005fran_005fugaussian_005ftail"></a>Function: <em>double</em> <strong>gsl_ran_ugaussian_tail</strong> <em>(const gsl_rng * <var>r</var>, double <var>a</var>)</em></dt>
<dt><a name="index-gsl_005fran_005fugaussian_005ftail_005fpdf"></a>Function: <em>double</em> <strong>gsl_ran_ugaussian_tail_pdf</strong> <em>(double <var>x</var>, double <var>a</var>)</em></dt>
<dd><p>These functions compute results for the tail of a unit Gaussian
distribution.  They are equivalent to the functions above with a standard
deviation of one, <var>sigma</var> = 1.
</p></dd></dl>





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