I attempted to solve for the definite integral for a binomial distribution:
I tried to solve through integration by parts but could not. In addition, I am not sure how to go past the factorial. How would I find this integral? Any help is appreciated, I am still relatively new to integration.
Let $X\sim\mathrm{Bin}(n,p)$. Then the distribution function of $X$ is $$F_X(x) := \mathbb P(X\leqslant x) = \sum_{k=0}^{\lfloor x\rfloor} \mathbb P(X=k). $$ This sum does not have a nice closed form; the best we can do is $$ \mathbb P(X\leqslant a) = \sum_{k=0}^{\lfloor a\rfloor} \binom nk p^k(1-p)^{n-k} = (n-a)\binom na\int_0^{1-p} t^{n-a-1}(1-t)^a\ \mathsf dt. $$ If $a\leqslant np$ then we can derive some upper bounds for the lower tail of the distribution function. Hoeffding's inequality yields $$ F_X(a)\leqslant \exp\left(-2\frac{(np-a)^2}n\right), $$ and Chernoff's inequality yields $$ F_X(a) \leqslant \exp\left(-\frac1{2p}\cdot\frac{(np-a)^2}n \right). $$