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 $qn = (1-p)n = n - np$:

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