Sum of the first $k$ terms used in computing the average of the binomial distribution I wonder if a "non-trivial" bound is known or could be computed (in closed form) for the sum of the first $k$ terms appearing in the computation of the average of the binomial distribution, i.e.,
$$\sum\limits_{i=0}^k \binom{N}{i} p^i (1-p)^{N-i}i ,$$ 
where $p$ is the probability of success, and $k < N$.
The obvious "trivial" bound is of course the actual value of the average which is $N p$, obtained for $k=N$.
Several bounds have been given for the "sum of the first $k$ binomial coefficients"
$f(N,k) = \sum_{i=0}^{k}\binom{N}{i}$ 
https://mathoverflow.net/questions/17202/sum-of-the-first-k-binomial-coefficients-for-fixed-n
if this is somehow of any help.
 A: Such sums can be bounded using concentration inequalities and limit theorems from the probability literature, and choosing which bound is most appropriate for the application at hand depends heavily on the asymptotic scaling of $k$ (as a function of $N$) as well as the desired degree of accuracy. The main three cases are when $k<(p-\epsilon)N$ (lower tail), $k=(p+o(1))N$ (Gaussian regime), and $k>(p+\epsilon)N$ (upper tail).
Even though it is the smallest region of the above three, I will assume you are interested in the Gaussian regime since it seems to be everybody's favorite. Using the same reasoning as in https://stats.stackexchange.com/questions/411164 one can deduce from the Central Limit Theorem that
$$
\lim_{n\to\infty}\sum_{i=0}^{k} \binom{n}{i}p^i(1-p)^{n-i}\frac{i-np}{\sqrt{np(1-p)}}=\frac{e^{-x^2/2}}{\sqrt{2\pi}},\qquad x=\lim_{n\to\infty}\frac{k-np}{\sqrt{np(1-p)}},
$$
which can be manipulated to give the difference between the weighted sums you are trying to bound, and the unweighted sums that are bounded more commonly.
Or perhaps you are not interested in the Gaussian regime at all, but some other regime like an "extreme" lower tail where $k=o(N)$, in which case any concentration inequality listed here, together with bounding $i$ by $k$ will get you a reasonably accurate bound.
