Let Z be a standard normal random variable. These notes present upper and lower bounds for the complementary cumulative distribution function
We prove simple bounds first then state improved bounds without proof.
An upper bound is easy to obtain. Since x/t > 1 for x in (t, ∞), we have
We can also show there is a lower bound
To prove this lower bound, define
We will show that g(t) is always positive. Clearly g(0) > 0. From the derivative
we see that g is strictly decreasing. Since the limit of g(t) as t goes infinity vanishes, g must always be positive.
Combining the inequalities above we have
Abramowitz and Stegun give bounds on the error function from which we can derive different bounds on the normal distribution. Formula 7.1.13 from Abramowitz and Stegun reads
Let t = √2x. Then the inequality above yields
This post includes Python code based on the inequality above and shows that the upper and lower bounds are quite close to each other as t gets large.