The question of bounding the tails of the normal distribution has popped up a couple of times on math.SE lately. This is an easy-to-prove but useful result, and so it’s worth talking about.
The standard normal probability density function is . We want to find a bound on for .
One simple way, since , is the following:
We can get improved bounds, though, by using repeated integration by parts. Ignoring the constant for now, introduce to the integrand. Then let and . Thus and . We get
Since the integral at the end is negative, we have the same bound as before: However, instead of stopping here we could continue to apply integration by parts to the integral on the right-hand side. For example, the next application (with and ) yields a lower bound:
(See, for example, Dilip Sarwaite’s answer here.) Continuing this process will produce successively better and better upper and lower bounds.
Added (January 23, 2012): Recently I found this nice technical report, “Bounding Standard Gaussian Tail Probabilities,” by Lutz Dümbgen. It surveys many known results on bounding the tails of the normal distribution and refines some of them. I modified one of the derivations in the report to prove an inequality that was asked about on math.SE recently.