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The question is:

Consider $n$ Bernoulli trials, where for $i = 1, 2,…, n$, the $i$th trial has probability $p_i$ of success, and let $X$ be the random variable denoting the total number of successes. Let $ p \ge p_i$ for all $i = 1, 2, \ldots , n$. Prove that for $ 1 \le k \le n$,

$$\Pr \{ X < k \} \ge \sum_{i=0}^{k-1}b(i; n, p)$$

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I tried to use induction on $k$ but obviously it doesn’t work.

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You’re trying to show that for $0\leq k\leq n$, $\mathbb{P}[X\leq k]\geq \mathbb{P}[\text{Bin}(n,p)\leq k]$.

We can prove this by coupling; the idea is to work on a probability space over which these variables are related in a useful way. Let $(U_i:1\leq i\leq n)$ be a sequence of IID uniform random variables on $[0,1]$ and write:

$$X=\sum\limits_{i=1}^n \mathbf{1}_{U_i<p_i}\quad\text{and}\quad

\text{Bin}(n,p) = \sum\limits_{i=1}^n \mathbf{1}_{U_i<p}$$

These both have the correct distributions.

Now $ X\leq \text{Bin}(n,p)$ with full probability. In particular, on this probability space, $\{\text{Bin}(n,p)\leq k\}\subset \{X\leq k\}$, so taking probabilities gives the result.

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