Mill's Inequality

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

Mill's Inequality — Let $Z\sim N(0,1)$ . Then^[1] $\operatorname {P} (|Z|>t)\leq {\sqrt {\frac {2}{\pi }}}{\frac {\exp(-t^{2}/2)}{t}}\leq {\frac {\exp(-t^{2}/2)}{t}}$

The looser bound shows the exponential shape. Compare this to the Chernoff bound:^[2]

$\operatorname {P} (|Z|>t)\leq 2\exp(-t^{2}/2)$

References

^ Wasserman, Larry (2004). "All of Statistics". Springer Texts in Statistics: 65. doi:10.1007/978-0-387-21736-9. ISSN 1431-875X.
^ Ma, Xuezhe. "Probability Inequalities 10/36-705 Intermediate Statistics Lecture Notes 2" (PDF).

[1] Wasserman, Larry (2004). "All of Statistics". Springer Texts in Statistics: 65. doi:10.1007/978-0-387-21736-9. ISSN 1431-875X.

[2] Ma, Xuezhe. "Probability Inequalities 10/36-705 Intermediate Statistics Lecture Notes 2" (PDF).

[1]

[2]