# What is the motivation for Gaussian Tail Bounds?

Perhaps I am missing something here, but I'm not seeing the value of having an upper tail bound for a Gaussian random variable. In my statistics class, we motivate tail inequalities for situations in which we have very limited information about a distribution yet we wish to understand the limitations of the uncertainty given our limited understanding of the distribution. E.g. Markov's inequality allows us to make statements about the probability of being a certain distance away from the (finite) mean of a non-negative random variable when all we know is the mean (and that the Random Variable is non-negative). Similarly for Chebychev's inequality: when all we know is the (finite) mean and variance of the distribution, we can still make statements about the probability of being a certain distance from the mean.

This is significantly different from when we talk about tail bounds for Gaussian random variables (or sums/averages of i.i.d. Gaussians): if we assume them to be Gaussian, then we know the entire PDF. Our original motivation for deriving a bound is now redundant -- why bound something when we can actually compute the probability directly using the PDF?

Perhaps there is value in the fact that computing such integrals can be difficult, and having a simpler (but less precise) bound allows for fast/simple approximations?

Let's say that you are trying to compute $\lim_{n \rightarrow \infty}\frac{1}{n} \log P(\bar{S}_n > t)$, where $\bar{S}_n$ is a sample mean of n i.i.d. Gaussians. If you know that the tail bound for a gaussian is approximately $e^{-\frac{t^2}{2 \sigma}}$ up to logarithmic factors, you can compute this value. Whereas, computing this from the integral can only give you numerical approximations.