I am trying to solve some bounds with Chernoff bounds.
My problem is that the values i get for $\delta$ are usually so large that it is not possible to compute what is the result.
As an example here is my problem:
I start with
$ P(X \geq (1+ \delta) \mu) < \left(\frac{e^\delta}{(1+\delta)^{(1+\delta)}}\right)^\mu $
Now I want to solve the probability of $X \geq 30$ but my $\mu$ is so small that I need $\delta$ equal to 2000 to reach my inequality but solving the Chernoff bound with such a $\delta$ would be impossible.
Now the problem is that I only have this formula of Chernoff bound as the other requires $0 \leq \delta \leq 1$.
Is there any way I can dodge this problem?
Best Answer
Well, it is not impossible to approximate:
https://www.wolframalpha.com/input/?i=e%5E2000+%2F+%281%2B2000%29%5E%281%2B2000%29+decimal+expansion
However, the probability is just ridiculously small. Since that particular Chernoff bound is for a sum of $n$ i.i.d. Bernoulli r.v.'s with mean $p$ (so that $\mu = np$) it might be the case that $n < 30$, so that, in fact, $\mathbb{P}(X\geq30) = 0$ and you don't need the bound at all.