Exercise C.4.8

$\star$ Consider $n$ Bernoulli trials, where $i = 1, 2, \ldots, n$, 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 $1 \le k \le n$,

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

Let's create another set of Bernoulli trials $Y$, each with probability $p$.

$$\Pr\{Y < k\} = 1 - \Pr\{Y \ge k\} \le 1 - \Pr\{X \ge k\} = \Pr\{X < k\}$$

The inequality follows from exercise C.4-9. That's right. We haven't done it yet, but we're using it. That's because I'm the Doctor!