Conjecture: Let $m$ and $n$ be fixed positive integers and let $f(k)$ be the probability that a Binomial($k(m+n)$, $p$) random variable is less than $kn$. Then for sufficiently small $p$, $f(k)$ is an increasing function of $k$.


Unless I've made a mistake, the conjecture holds if you replace the binomial with its normal approximation and $p < n/(m+n)$. This suggests the conjecture should be true for large $m$ and $n$. But is it true for all $m$ and $n$, i.e. for small values? How large can $p$ be?



Binomial distribution conjecture
Unless I've made a mistake: the conjecture is true for $p<kn/(kn+km+1)$ and possibly other values as well.


Requiring $k(m+n)$ to be an integer, $f(k) = \displaystyle\sum\_0^{\lfloor kn\rfloor}(km+kn)Cr p^r (1-p)^km+kn-r$ and a similar expression with the same limits gives a lower bound for $f(k +1/(m+n))$. Considering the ratio of equivalent terms in this sum: if this is greater than $1$, $f(k + 1/[m+n])>f(k)$ which happens if $p<kn/(km+kn+1)$.
Hope this helps!


