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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.

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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 $$p'-p = 0.25 < q'-q = 0.3$$

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