# Properties of cumulative binomial distribution

Let $F\left(k, n, p\right) = \sum_{i=1}^k\binom{n}{i}p^i\left(1-p\right)^{n-i}$ denote the cumulative binomial distribution function.

If

$F\left(k, n, p\right)-F\left(k, n, p'\right) \geq F\left(k, n, q\right) - F\left(k, n, q'\right) > 0$,

does this imply that

$p'-p \geq q'-q$

I think that this should hold, but I have spent a week trying to prove it without success.

Any help is appreciated.

There are counterexamples (At least for the definitions of binomial CDF from Maple or Wikipedia, where the sum starts at $i=0.$ So please check with your values.)
With $n=10, k=5, p=0.5, p'=0.75, q=0.2, q'=0.5\;$ I compute $$F(k,n,p)=F(k,n,q')=0.623046875\\ F(k,n,p')=0.0781269073e-1\\ F(k,n,q)= 0.9936306176$$ And therefore $$F(k, n, p)-F(k, n, p') = 0.54492 \geq F(k, n, q) - F(k, n, q') =0.37058 > 0,$$ but $$p'-p = 0.25 < q'-q = 0.3$$