Proving an inequation based on binomial distributions

Question

Problem statement

Let $c \in \mathbb{N}$, $n_1 \in \mathbb{N}_0$, and $n_2 \in \mathbb{N}_0$ be integers and $p$ a probability. Furthermore, let $b(m,j,p) = \binom{m}{j}p^j(1-p)^{m-j}$ denote the probability mass function of the binomial distribution and let $B(m,j,p) = \sum_{i=0}^j b(m,i,p) = \sum_{i=0}^j \binom{m}{i}p^i(1-p)^{m-i}$ denote the cumulative binomial distribution function.

I want to prove that

$\forall k \in \mathbb{N}, k \leq c: \sum_{j=0}^{k-1} b\left(c,j,p\right) B_{j,k} \leq \sum_{j=0}^{k-1} b\left(c-1,j,p\right) B_{j,k}$

holds when $B_{j,k}$ is defined as

$B_{j,k} = B\left(n_1,k-1-j,p\right) \cdot B\left(n_2,k-1-j,p\right)$

What I have done so far

For an arbitrary $k \in \mathbb{N}$ satisfying $k \leq c$, we have \begin{align} &\sum_{j=0}^{k-1} b\left(c,j,p\right) B_{j,k} \leq\sum_{j=0}^{k-1} b\left(c-1,j,p\right) B_{j,k}\\ \iff \; &\sum_{j=0}^{k-1} \binom{c}{j}p^j(1-p)^{c-j} B_{j,k} \leq \sum_{j=0}^{k-1} \binom{c-1}{j}p^j(1-p)^{c-j-1} B_{j,k}\\ \iff \; &\sum_{j=0}^{k-1} \binom{c}{j}p^j(1-p)^{c-j} B_{j,k} \leq \frac{\sum_{j=0}^{k-1} \binom{c-1}{j}p^j(1-p)^{c-j} B_{j,k}}{1-p}\\ \iff \; &\sum_{j=0}^{k-1} (1-p)\binom{c}{j}p^j(1-p)^{c-j} B_{j,k} \leq \sum_{j=0}^{k-1} \binom{c-1}{j}p^j(1-p)^{c-j} B_{j,k}\nonumber\\ \iff \; &\sum_{j=0}^{k-1} \left[\binom{c-1}{j}-(1-p)\binom{c}{j}\right] p^j(1-p)^{c-j} B_{j,k} \geq 0\nonumber\\ \iff \; &\sum_{j=0}^{k-1} \left[\frac{c-j}{c}\binom{c}{j}-(1-p)\binom{c}{j}\right] p^j(1-p)^{c-j} B_{j,k} \geq 0\nonumber\\ \iff \; &\sum_{j=0}^{k-1} \frac{cp-j}{c} \binom{c}{j} p^j(1-p)^{c-j} B_{j,k} \geq 0\nonumber\\ \iff \; &\sum_{j=0}^{k-1} (cp-j) \binom{c}{j} p^j(1-p)^{c-j} B_{j,k} \geq 0 \end{align}

Unfortunately, at this point, I'm unsure how to continue, and would greatly appreciate any help!

Here are a few observations which might be useful:

• Because of the factor $cp-j$, every summand in in the last inequation with $j \leq \lfloor cp \rfloor$ is non-negative, while every summand with $j > \lfloor cp \rfloor$ is negative.
• The factors $B_{j,k}$ are monotonically decreasing with increasing $j$ and thus reduce the effect of the "harmful" negative summands. However, experimental evaluation suggests that the last inequation could be satisfied even in the absence of the $B_{j,k}$ factors. Maybe this could be exploited.

Answer 1

Without the $B_{j,k}$, the last sum can be evaluated: $$ \sum_{j=0}^{k-1} (cp-j) \binom cj p^j (1-p)^{c-j} = k\binom ck p^k(1-p)^{c-k+1}. $$ Together with your observation that $B_{j,k}$ is decreasing, that does it, no?