(i) Suppose X has a binomial distribution with probability of success p over n trials. Show that

$\displaystyle E(X) \left[\frac{1}{1 + X}\right] = \frac{1 - (1-p)^{n+1}}{p(n+1)}$

(ii) Suppose now X has a negative binomial distribution with density

$\displaystyle P(X=k) = \left(\begin{array}{cc}k-1\\n-1\end{array}\right) p^nq^{k-n}$

with $\displaystyle q = 1-p$ and $\displaystyle k \geq n$. Prove that for any function f(x),

$\displaystyle E[qf(X)] = E \left[\frac{(X-n)f(X-1)}{X-1}\right]$