# The Gaussian Tail Distribution¶

gsl_ran_gaussian_tail(a, 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,

$N(a;\sigma) = (1/2) \operatorname{erfc}(a / \sqrt{2 \sigma^2}).$
gsl_ran_gaussian_tail_pdf(x, a, 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.

gsl_ran_ugaussian_tail(a)
gsl_ran_ugaussian_tail_pdf(x, 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.