Exercise C.5.2

$\star$ Prove corollaries C.6 and C.7

Let $Y$ be a random variable, indicating the number of failures.

For C.6 (using C.4):

$$ \Pr\{X > k\} = \Pr\{Y < n-k\} < \frac{(n-k)p}{nq - n + k}b(n-k;n,q) = \frac{(n-k)p}{k-np}\binom{n}{n-k}q^{n-k}p^k = \frac{(n-k)p}{k-np}b(k;n,p) $$

For C.7 (using C.5):

$$ \Pr\{Y < n - k\} < \frac{1}{2} \Pr\{Y < n - k + 1\} \\ \Downarrow \\ \Pr\{X > k\} < \frac{1}{2} \Pr\{X > k - 1\} $$