# Exercise C.5.5

$\star$ Show that the conditions of theorem C.8 imply that

$$\Pr\{\mu - X \ge r\} \le \bigg(\frac{(n - \mu)e}{r}\bigg)^r$$

Similarly, show that the conditions of corollary C.9 imply that

$$\Pr\{np - X \ge r\} \le \bigg(\frac{nqe}{r}\bigg)^r$$

This is tricky. Let's introduce a new random variable $Y = n - X$.

$$\nu = E[Y] = E[n - X] = n - E[x] = n - \mu$$

Using theorem C.8, we get:

$$\Pr\{Y - \nu > r\} \le \bigg(\frac{\nu e}{r}\bigg)^r \\ \Downarrow \\ \Pr\{\mu - X \ge r\} \le \bigg(\frac{(n - \mu)e}{r}\bigg)^r$$

It's similar with the other one, where $qn = (1-p)n = n - np$:

$$\Pr\{np - X\} = \Pr\{n - X - n + np\} = \Pr\{Y - qn > r\} \le \bigg(\frac{nqe}{r}\bigg)^r$$