I have the following binomial distribution series
$$ f(n,p) = \sum_{k=n+1}^{2n}\binom{2n}{k}p^k (1-p)^{2n-k} $$
I want to show the following properties:
- If $p>1/2$, $f(n,p)\rightarrow 1$ as $n\gg 1$
- What is the value of $p<1/2$ for which $f(n,p)$ has a maximum at $n\geq 2$
How can I do it without explicitly computing the series?
For the first point, I have a non-rigorous argument. Notice that I can express the series as $$ f(n,p) = 1-\sum_{k=0}^{n}\binom{2n}{k}p^k (1-p)^{2n-k} $$
The maximum of the binomial distribution is $k=2 n p$, therefore each term of the series grows as $k$ grows, if $p>1/2$. Let me estimate the biggest term for $k=n$ at large $n$ as
$$ \binom{2n}{k}p^k (1-p)^{2n-k}\Big|_{k=n} \sim \frac{(2n)^n}{n^n}(1-p)^{2n}\left(\frac{p}{1-p}\right)^{n} = \left[2p(1-p)\right]^n \rightarrow 0 $$
since the term inside the brackets is always less than 1. Would this argument be ok? What about the second point?