Probability functions - GNU Scientific Library -- Reference Manual

The probability functions for the Normal or Gaussian distribution are described in Abramowitz & Stegun, Section 26.2.

— Function: double gsl_sf_erf_Z (double x)
— Function: int gsl_sf_erf_Z_e (double x, gsl_sf_result * result)

These routines compute the Gaussian probability density function Z(x) = (1/\sqrt{2\pi}) \exp(-x^2/2).

— Function: double gsl_sf_erf_Q (double x)
— Function: int gsl_sf_erf_Q_e (double x, gsl_sf_result * result)

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).

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) / \erfc(x/\sqrt 2)

It decreases rapidly as x approaches -\infty and asymptotes to h(x) \sim x as x approaches +\infty.

— Function: double gsl_sf_hazard (double x)
— Function: int gsl_sf_hazard_e (double x, gsl_sf_result * result)

These routines compute the hazard function for the normal distribution.