GNU Scientific Library – Reference Manual: The Gaussian Tail Distribution

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### 20.3 The Gaussian Tail Distribution

Function: double gsl_ran_gaussian_tail (const gsl_rng * r, double a, double sigma)

  This function provides random variates from the upper tail of a Gaussian distribution with standard deviation sigma. The values returned are larger than the lower limit a, which must be positive. The method is based on Marsaglia’s famous rectangle-wedge-tail algorithm (Ann. Math. Stat. 32, 894–899 (1961)), with this aspect explained in Knuth, v2, 3rd ed, p139,586 (exercise 11).

  The probability distribution for Gaussian tail random variates is,

      p(x) dx = {1 \over N(a;\sigma) \sqrt{2 \pi \sigma^2}} \exp (- x^2/(2 \sigma^2)) dx

  for x > a where N(a;\sigma) is the normalization constant,

      N(a;\sigma) = (1/2) erfc(a / sqrt(2 sigma^2)).

Function: double gsl_ran_gaussian_tail_pdf (double x, double a, double sigma)

  This function computes the probability density p(x) at x for a Gaussian tail distribution with standard deviation sigma and lower limit a, using the formula given above.

Function: double gsl_ran_ugaussian_tail (const gsl_rng * r, double a)
Function: double gsl_ran_ugaussian_tail_pdf (double x, double a)

  These functions compute results for the tail of a unit Gaussian distribution. They are equivalent to the functions above with a standard deviation of one, sigma = 1.