# Math Help - Binomial Distribution Proving.

1. ## Binomial Distribution Proving.

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

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

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

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

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

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

2. For the first one I would rewrite ${X\over X+1}={X+1-1\over X+1}=1-{1\over X+1}$

and obtain $1-E\biggl({1\over X+1}\biggr)$ instead.

That can be done by substituting y=x+1 in your sum.

3. Originally Posted by Moo