Mill's Inequality

Mill's Inequality is a useful concentration/tail bound on Normally distributed random variables.

Let $Z\sim N(0,1)$. Then $$\operatorname {P} (|Z|>t)\leq {\sqrt {\frac {2}{\pi }}}{\frac {\exp(-t^{2}/2)}{t}}$$Equivalently, shift and rescale $Z\sim N(\mu ,\sigma ^{2})$. Then$$\operatorname {P} (|Z-\mu |>t)\leq {\sqrt {\frac {2}{\pi }}}{\frac {\exp(-t^{2}/2\sigma ^{2})}{t}}$$