A General Method for the Approximation of Tail Areas

For a density function $f(x)$, the tail area α(x) = ∫^∞ₓ f(x) dx, may be approximated by α̂(x) = f(x)/g(x) · (K - 1)⁻¹·\1 + 1/2 (g'(x)/g²(x) - (K))\, where $g(x) = f(x)/f'(x)$, and K = limx→∞ '(x)/g²(x)\. The formula requires only one constant and three function evaluations; g and $g'$ are typically elementary functions. Such approximations are useful for programmed calculators or very small computers where only a few constants can be stored. The accuracy of the approximation is calculated for some common distributions. The approximation is very accurate for a large class of distributions.