Probability functions
The probability functions for the Normal or Gaussian distribution are
described in Abramowitz & Stegun, Section 26.2.
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gsl_sf_erf_Z(x)
This routine computes the Gaussian probability density function
\(Z(x) = (1/\sqrt{2\pi}) \exp(-x^2/2)\).
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gsl_sf_erf_Q(x)
This routine computes the upper tail of the Gaussian probability
function \(Q(x) = (1/\sqrt{2\pi}) \int_x^\infty \exp(-t^2/2) dt\).
The hazard function for the normal distribution, also known as the
inverse Mills’ ratio, is defined as,
\[h(x) = Z(x)/Q(x) =
\sqrt{2/\pi} \exp(-x^2 / 2) / \operatorname{erfc}(x/\sqrt 2)\]
It decreases rapidly as \(x\) approaches \(-\infty\) and asymptotes
to \(h(x) \sim x\) as \(x\) approaches \(+\infty\).
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gsl_sf_hazard(x)
This routine computes the hazard function for the normal distribution.