A way to possibly calculate one Binomial CDF function from another closely related one?

Question

Let $y < z$ be two numbers between $0$ and $1$, is there a way to relate the CDF functions $F_{n,y}(s)$ and $F_{n,z}(s)$... or approximate one from another, without just saying $F_{n,z}(s) \le F_{n,y}(s)$ for all $s$?

Answer 1

Even though there is no closed form for the CDF of the binomial distribution, there is one for the derivative with respect to the $p$ parameter.

Namely, if $$ F = \sum_{i=0}^s \binom ni p^i(1-p)^{n-i}, $$ then $$ \frac{dF}{dp} = -(n-s)p^s(1-p)^{n-s-1}\binom ns. $$

In particular, in the notation of your question, $$ F_{n,y}(s)-F_{n,z}(s) = (n-s)\binom ns \int_y^z p^s(1-p)^{n-s-1}\,dp. $$

Answer 2

Under the conditions where the Central Limit Theorem applies ($n \to \infty$ with $p$ approaching a constant), you can approximate the binomial CDF by $$ F_{n,p}(s) \approx \Phi\left(\dfrac{s-np}{\sqrt{np(1-p)}}\right) $$ where $\Phi$ is the standard normal CDF. We should be able to differentiate this with respect to $p$, obtaining $$ \eqalign{F_{n,z}(s) - F_{n,y}(s) &\approx (z-y) \dfrac{\partial}{\partial p} \left.\Phi\left(\dfrac{s-np}{\sqrt{np(1-p)}}\right)\right|_{p=y}\cr & = \dfrac{(z-y)((2y-1)s - ny)}{2\sqrt{2\pi n}\; y^{3/2} (1-y)^{3/2}} \exp\left(- \dfrac{(s-ny)^2}{2ny(1-y)}\right)\cr}$$