## File: Probability-functions.html

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 123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113  GNU Scientific Library – Reference Manual: Probability functions

7.15.4 Probability functions

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.