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<title>GNU Scientific Library – Reference Manual: Probability functions</title>
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<a name="Probability-functions"></a>
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<p>
Previous: <a href="Log-Complementary-Error-Function.html#Log-Complementary-Error-Function" accesskey="p" rel="previous">Log Complementary Error Function</a>, Up: <a href="Error-Functions.html#Error-Functions" accesskey="u" rel="up">Error Functions</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<a name="Probability-functions-1"></a>
<h4 class="subsection">7.15.4 Probability functions</h4>
<p>The probability functions for the Normal or Gaussian distribution are
described in Abramowitz & Stegun, Section 26.2.
</p>
<dl>
<dt><a name="index-gsl_005fsf_005ferf_005fZ"></a>Function: <em>double</em> <strong>gsl_sf_erf_Z</strong> <em>(double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005ferf_005fZ_005fe"></a>Function: <em>int</em> <strong>gsl_sf_erf_Z_e</strong> <em>(double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These routines compute the Gaussian probability density function
<em>Z(x) = (1/\sqrt{2\pi}) \exp(-x^2/2)</em>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005fsf_005ferf_005fQ"></a>Function: <em>double</em> <strong>gsl_sf_erf_Q</strong> <em>(double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005ferf_005fQ_005fe"></a>Function: <em>int</em> <strong>gsl_sf_erf_Q_e</strong> <em>(double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These routines compute the upper tail of the Gaussian probability
function
<em>Q(x) = (1/\sqrt{2\pi}) \int_x^\infty dt \exp(-t^2/2)</em>.
</p></dd></dl>
<a name="index-hazard-function_002c-normal-distribution"></a>
<a name="index-Mills_0027-ratio_002c-inverse"></a>
<p>The <em>hazard function</em> for the normal distribution,
also known as the inverse Mills’ ratio, is defined as,
</p>
<div class="example">
<pre class="example">h(x) = Z(x)/Q(x) = \sqrt{2/\pi} \exp(-x^2 / 2) / \erfc(x/\sqrt 2)
</pre></div>
<p>It decreases rapidly as <em>x</em> approaches <em>-\infty</em> and asymptotes
to <em>h(x) \sim x</em> as <em>x</em> approaches <em>+\infty</em>.
</p>
<dl>
<dt><a name="index-gsl_005fsf_005fhazard"></a>Function: <em>double</em> <strong>gsl_sf_hazard</strong> <em>(double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005fhazard_005fe"></a>Function: <em>int</em> <strong>gsl_sf_hazard_e</strong> <em>(double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These routines compute the hazard function for the normal distribution.
</p></dd></dl>
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