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_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.