How to find a lower bound of the probability of a Gaussian random variable

Question

I want to find the lower bound of $$\int_{-a}^{a}e^{-\frac{x^2}{2\sigma^2}}\frac{dx}{\sqrt{2\pi }\sigma}$$ where $x$ is a Gaussian Random variable with mean $0$ and variance $\sigma^2$.

I notice there are bunch of way to find a bound for standard Gaussian's tail probability but how should we deal with the probability in the middle?

Thanks in advance!

Answer 1

I presume $x$ and $y$ are the same variable, so this is $\Phi(a/\sigma) - \Phi(-a/\sigma) = 2 \Phi(a/\sigma)-1 = \text{erf}(a/(\sqrt{2}\sigma))$, where $\Phi$ is the standard normal CDF.

One lower bound that is good for $a/\sigma$ near $0$ is $$ \sqrt{\frac{2}{\pi}} \left(\frac{a}{\sigma} - \frac{a^3}{6\sigma^3}\right) $$ Of course this is useless for $a > \sqrt{6} \sigma$.

Answer 2

Another lower bound, as derived in this answer, is $$ \begin{align} \int_{-a}^ae^{-\frac{x^2}{2\sigma^2}}\frac{\mathrm{d}x}{\sqrt{2\pi}\,\sigma} &=\sqrt{\frac2\pi}\int_0^{a/\sigma}e^{-x^2/2}\,\mathrm{d}x\\ &=\sqrt{\frac2\pi}e^{-\frac{a^2}{2\sigma^2}}\left(\frac{a}\sigma+\frac{a^3}{3\sigma^3}\right) \end{align} $$ This works as a lower bound for all $a$. To be useful for larger $a$, we need to use more terms of the series $$ \sqrt{\frac2\pi}e^{-\frac{a^2}{2\sigma^2}}\sum_{k=0}^\infty\frac{a^{2k+1}}{(2k+1)!!\,\sigma^{2k+1}} $$