Mills ratio

For the standard normal density $\varphi$ and distribution function $\Phi$, the upper-tail Mills ratio is $(1-\Phi (x))/\varphi (x)$. Ratios involving a Gaussian density and tail probability control truncated-normal moments and Gaussian-tail asymptotics.

Two successive integrations by parts of the standard normal density tail give positive remainder integrals establishing these strict inequalities for every $x>0$. The lower bound can be negative at small $x$ without invalidating the inequality.

An inverse Mills ratio divides a Gaussian density by a Gaussian tail probability. For lower truncation at zero, the conditional mean correction is $\sigma \varphi (\mu /\sigma )/\Phi (\mu /\sigma )$.