Q: How can I approximate the sum of a binomial probability times the sum's index? Is there a way to approximate $f(b)$ / remove the summation? All variables are constant except for $i$. $c$ is a probability between 0 - 1 and $a<b$. $P_i$ is a binomial probability.
$$
f(b)=\sum_{i=a}^{b}P_ii
$$
$$
P_i=\binom{b}{i}c^i(1-c)^{b-i}
$$

A: Are you sure that your definition of your term $P_i$ is right? Because if it is the case, then $P_i$ doesn't actually depend on $i$, and your sum $f(b)$ just becomes:
$f(b) = \sum_{i=a}^b P_i i = \binom{b}{a}c^a(1-c)^{b-a} \sum_{i=a}^b i = \binom{b}{a}c^a(1-c)^{b-a} \frac{1}{2}(b(b+1)-(a-1)a)$
(using the Summation formula of Gauss)
I can imagine that you $P_i$ denotes actually $P_i = \binom{b}{i}c^i(1-c)^{b-i}$, you need to tell me :)
