Mill's Inequality Explained

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

Let

Z \sim N( 0 , 1 ) . Then^[1] $$\backslash operatorname\; (|Z|\; >\; t)\; \backslash le\; \backslash sqrt\; \backslash frac$$Equivalently, shift and rescale Z \sim N(\mu, \sigma ^{2)} . Then^[2] $$\backslash operatorname\; (|Z\; -\; \backslash mu|\; >\; t)\; \backslash le\; \backslash sqrt\; \backslash frac$$

References

1. Wasserman . Larry . 2004 . All of Statistics . Springer Texts in Statistics . 65 . 10.1007/978-0-387-21736-9 . 1431-875X.
2. Web site: Sub-Gaussian Random Variables . MIT OCW.