tail estimation of normal distribution

Question

In https://www.johndcook.com/blog/2021/11/05/normal-tail-estimate/ , why "base of our rectangle runs from x to the point t where the integrand drops by a half" ?

Answer 1

FWHM is a very simple approximation for taking an integral of a function that, I believe, was popular for analyzing spectogram images (where you would have a curve with peaks and would need to estimate the area under those peaks by hand). Roughly speaking, if $f$ is a (probably fairly smooth) non-negative function on an interval $[a, b]$ with:

• $f(a) = f(b) = 0$
• $\max_{[a, b]} f(x) = M$ and
• $f(c) = f(d) = \frac{M}{2}$

then $\int_a^b f(x) dx \approx M|d - c|$. If you look at the image below, the grey rectangle is bounded by the $x$ axis, the maximum of the function, and two points where $f$ is half of its maximum. The FWHM heuristic assumes that the area of that rectangle is approximately the same as the integral of the function over the whole domain, which is the same as assuming that the part of the rectangle that sits above the curve has approximately the same area as the part of the curve that sits outside the rectangle.

In the blog post you linked, he's using a one-sided variation of the same trick, but the same general idea applies - find the value of $t$ for which $f(t) = \frac{1}{2}\max f(x)$, and assume that the integral can be approximated by a rectangle of height $\max f(x)$ and width $t - x_0$.

Unfortunately, other than this blog, I couldn't find anything discussing the accuracy of the approximation or the conditions under which it might perform well