I recently met a problem and I have been finding related materials about it. However it is still hard to handle.
Precisely, I want to maximize a function related to $$S_n = \sum_{i=0}^{\lfloor \frac{n}{2}\rfloor} \binom{n}{i}p^i(1-p)^{n-i}$$, i.e., the probability of a minority event.
I then tried with Chernoff bound and assume it has an approximation as $$S_n \sim e^{-\lambda n}$$, and this works well and leads to the result that the function achieves its max value at some $x$.
But actually the original function may have a different shape, and even asymptotically the peak points can be different.
So is there a better way to bound the sum and get the remainder items?
Thanks in advance!