I'm looking for a lower bound for binomial random variable $X$~ $B(n,p)$, where $p=(1+\epsilon)/2$ for $\epsilon >0$. I want to bound $Pr(X> n/2)$.
I know Suld's inequality, but it is good for the case $p=(1-\epsilon)/2$.
The best I could get is :
$Pr(X> n/2) =$ $\sum_{k=n/2}^{n}$ $ {n}\choose{k}$$p^k(1-p)^{n-k}$ > $\sum_{k=n/2}^{n}$ $ {n}\choose{k}$$(1/2)^{n}$$=1/2$.
Which is not so good for me.
Is there any tighter bound?
Thanks!