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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.