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How can I prove that the following function $f(p)$ is non-increasing in $p$: \begin{align*} f(p)=\sum_{i=a}^{N}\left(1-\frac{1}{b \cdot(i-1)}\right)\binom{N}{i}(1-p)^ip^{N-i} \end{align*} where $N$ and $a$ ($2\le a \le N$) are integer constants and $b$ is a positive real constant? I verify this by plotting the function for certain values.

Note: Without the first coefficient $\left(1-\frac{1}{b(i-1)}\right)$, which does not depend on $p$, the function is close to cdf of binomial distribution.

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OK, I found out how it works. I should have also said $b>\frac{1}{N-1}$ (i.e. the 'coefficients' are also all positive). We make use of the fact that $1-\frac{1}{b(i-1)}<1$ and does not depend on $p$. Taking the first derivative in $p$ gives that is smaller than the first derivative of a sum which is 1-CDF of a binomial distribution $B(1-p,N)$. This function is decreasing, hence the first derivative is $<0$. Finally, $f'(p)<0$.

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