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<a name="The-Gaussian-Distribution"></a>
<div class="header">
<p>
Next: <a href="The-Gaussian-Tail-Distribution.html#The-Gaussian-Tail-Distribution" accesskey="n" rel="next">The Gaussian Tail Distribution</a>, Previous: <a href="Random-Number-Distribution-Introduction.html#Random-Number-Distribution-Introduction" accesskey="p" rel="previous">Random Number Distribution Introduction</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-Distribution-1"></a>
<h3 class="section">20.2 The Gaussian Distribution</h3>
<dl>
<dt><a name="index-gsl_005fran_005fgaussian"></a>Function: <em>double</em> <strong>gsl_ran_gaussian</strong> <em>(const gsl_rng * <var>r</var>, double <var>sigma</var>)</em></dt>
<dd><a name="index-Gaussian-distribution"></a>
<p>This function returns a Gaussian random variate, with mean zero and
standard deviation <var>sigma</var>.  The probability distribution for
Gaussian random variates is,
</p>
<div class="example">
<pre class="example">p(x) dx = {1 \over \sqrt{2 \pi \sigma^2}} \exp (-x^2 / 2\sigma^2) dx
</pre></div>

<p>for <em>x</em> in the range <em>-\infty</em> to <em>+\infty</em>.  Use the
transformation <em>z = \mu + x</em> on the numbers returned by
<code>gsl_ran_gaussian</code> to obtain a Gaussian distribution with mean
<em>\mu</em>.  This function uses the Box-Muller algorithm which requires two
calls to the random number generator <var>r</var>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fran_005fgaussian_005fpdf"></a>Function: <em>double</em> <strong>gsl_ran_gaussian_pdf</strong> <em>(double <var>x</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 distribution with standard deviation <var>sigma</var>, using
the formula given above.
</p></dd></dl>

<br>

<dl>
<dt><a name="index-gsl_005fran_005fgaussian_005fziggurat"></a>Function: <em>double</em> <strong>gsl_ran_gaussian_ziggurat</strong> <em>(const gsl_rng * <var>r</var>, double <var>sigma</var>)</em></dt>
<dt><a name="index-gsl_005fran_005fgaussian_005fratio_005fmethod"></a>Function: <em>double</em> <strong>gsl_ran_gaussian_ratio_method</strong> <em>(const gsl_rng * <var>r</var>, double <var>sigma</var>)</em></dt>
<dd><a name="index-Ziggurat-method"></a>
<p>This function computes a Gaussian random variate using the alternative
Marsaglia-Tsang ziggurat and Kinderman-Monahan-Leva ratio methods.  The
Ziggurat algorithm is the fastest available algorithm in most cases.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fran_005fugaussian"></a>Function: <em>double</em> <strong>gsl_ran_ugaussian</strong> <em>(const gsl_rng * <var>r</var>)</em></dt>
<dt><a name="index-gsl_005fran_005fugaussian_005fpdf"></a>Function: <em>double</em> <strong>gsl_ran_ugaussian_pdf</strong> <em>(double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fran_005fugaussian_005fratio_005fmethod"></a>Function: <em>double</em> <strong>gsl_ran_ugaussian_ratio_method</strong> <em>(const gsl_rng * <var>r</var>)</em></dt>
<dd><p>These functions compute results for the unit Gaussian distribution.  They
are equivalent to the functions above with a standard deviation of one,
<var>sigma</var> = 1.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fcdf_005fgaussian_005fP"></a>Function: <em>double</em> <strong>gsl_cdf_gaussian_P</strong> <em>(double <var>x</var>, double <var>sigma</var>)</em></dt>
<dt><a name="index-gsl_005fcdf_005fgaussian_005fQ"></a>Function: <em>double</em> <strong>gsl_cdf_gaussian_Q</strong> <em>(double <var>x</var>, double <var>sigma</var>)</em></dt>
<dt><a name="index-gsl_005fcdf_005fgaussian_005fPinv"></a>Function: <em>double</em> <strong>gsl_cdf_gaussian_Pinv</strong> <em>(double <var>P</var>, double <var>sigma</var>)</em></dt>
<dt><a name="index-gsl_005fcdf_005fgaussian_005fQinv"></a>Function: <em>double</em> <strong>gsl_cdf_gaussian_Qinv</strong> <em>(double <var>Q</var>, double <var>sigma</var>)</em></dt>
<dd><p>These functions compute the cumulative distribution functions
<em>P(x)</em>, <em>Q(x)</em> and their inverses for the Gaussian
distribution with standard deviation <var>sigma</var>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fcdf_005fugaussian_005fP"></a>Function: <em>double</em> <strong>gsl_cdf_ugaussian_P</strong> <em>(double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fcdf_005fugaussian_005fQ"></a>Function: <em>double</em> <strong>gsl_cdf_ugaussian_Q</strong> <em>(double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fcdf_005fugaussian_005fPinv"></a>Function: <em>double</em> <strong>gsl_cdf_ugaussian_Pinv</strong> <em>(double <var>P</var>)</em></dt>
<dt><a name="index-gsl_005fcdf_005fugaussian_005fQinv"></a>Function: <em>double</em> <strong>gsl_cdf_ugaussian_Qinv</strong> <em>(double <var>Q</var>)</em></dt>
<dd><p>These functions compute the cumulative distribution functions
<em>P(x)</em>, <em>Q(x)</em> and their inverses for the unit Gaussian
distribution.
</p></dd></dl>

<hr>
<div class="header">
<p>
Next: <a href="The-Gaussian-Tail-Distribution.html#The-Gaussian-Tail-Distribution" accesskey="n" rel="next">The Gaussian Tail Distribution</a>, Previous: <a href="Random-Number-Distribution-Introduction.html#Random-Number-Distribution-Introduction" accesskey="p" rel="previous">Random Number Distribution Introduction</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>
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