Let \(X \sim \mathcal{N}(0, 1)\) and \(x > 0\).

\(

\mathbb{P}(X > x)

\)

\(

= \frac{1}{\sqrt{2\pi}} \int_x^\infty e^{-t^2/2} \,\mathrm{d}t

\)

\(

\leq \frac{1}{\sqrt{2\pi}} \int_x^\infty \frac{t}{x} e^{-t^2/2} \,\mathrm{d}t

\)

\(

= \frac{e^{-x^2/2}}{x \sqrt{2\pi}}.

\)

Quite elegant!

0-fold cross-validation@0foldcv@mathstodon.xyzAnother elegant bound that's stronger for small x is

P(|X| > x) ≤ exp(-x² / 2)

for X ~ N(0, 1) and x >0.

(but I'm not aware of a proof quite as short as the previous one)