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<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.
<div class="defun">
— Function: double <b>gsl_sf_erf_Z</b> (<var>double x</var>)<var><a name="index-gsl_005fsf_005ferf_005fZ-478"></a></var><br>
— Function: int <b>gsl_sf_erf_Z_e</b> (<var>double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005ferf_005fZ_005fe-479"></a></var><br>
<blockquote><p>These routines compute the Gaussian probability density function
<!-- {$Z(x) = (1/\sqrt{2\pi}) \exp(-x^2/2)$} -->
Z(x) = (1/\sqrt{2\pi}) \exp(-x^2/2).
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_sf_erf_Q</b> (<var>double x</var>)<var><a name="index-gsl_005fsf_005ferf_005fQ-480"></a></var><br>
— Function: int <b>gsl_sf_erf_Q_e</b> (<var>double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005ferf_005fQ_005fe-481"></a></var><br>
<blockquote><p>These routines compute the upper tail of the Gaussian probability
function
<!-- {$Q(x) = (1/\sqrt{2\pi}) \int_x^\infty dt \exp(-t^2/2)$} -->
Q(x) = (1/\sqrt{2\pi}) \int_x^\infty dt \exp(-t^2/2).
<!-- Exceptional Return Values: none -->
</p></blockquote></div>
<p><a name="index-hazard-function_002c-normal-distribution-482"></a><a name="index-Mill_0027s-ratio_002c-inverse-483"></a>The <dfn>hazard function</dfn> for the normal distribution,
also known as the inverse Mill's ratio, is defined as,
<pre class="example"> h(x) = Z(x)/Q(x) = \sqrt{2/\pi} \exp(-x^2 / 2) / \erfc(x/\sqrt 2)
</pre>
<p class="noindent">It decreases rapidly as x approaches -\infty and asymptotes
to h(x) \sim x as x approaches +\infty.
<div class="defun">
— Function: double <b>gsl_sf_hazard</b> (<var>double x</var>)<var><a name="index-gsl_005fsf_005fhazard-484"></a></var><br>
— Function: int <b>gsl_sf_hazard_e</b> (<var>double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005fhazard_005fe-485"></a></var><br>
<blockquote><p>These routines compute the hazard function for the normal distribution.
<!-- Exceptional Return Values: GSL_EUNDRFLW -->
</p></blockquote></div>
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