Find the expected value of $\frac{1}{X+1}$ where $X$ is binomial The problem:
X is a binomial random variable, find $E[\frac{1}{X+1}]$
n and p are not given
PDF for a binomial distribution is $\binom{n}{k}p^k(1-p)^{n-k}$
Expected value is
$\sum{x_ip(x_i)}$
But this is where I get stuck, I'm really rusty on my statistics and I'm not sure exactly how to structure it in the next step? I think I want to get the form of the following out of the summation
$\sum _{k=0}^{n} \binom{n}{k}p^k(1-p)^{n-k} = (p + 1 - p)^n = 1$
But I'm not sure if it should look like
$\sum \frac{1}{xp(x)+1} $
and if it should where to go from here?
 A: Another way to solve:
Use the fact $E\left[ \dfrac{1}{X+a}\right]=\int_{0}^{1}t^{a-1}. P_X(t)\, dt$
where $P_X(t)$ is PGF(Probability Generating Function) .
So 
$$\begin{align} E\left[ \dfrac{1}{X+1}\right]&=\int_{0}^{1}t^0. P_X(t)\, dt \\ &=\int_{0}^{1} (q+pt)^n\, dt \\ &=\dfrac{1-q^{n+1}}{(n+1)p}\end{align}$$ where $q=1-p$.
A: Using the law of the unconscious statistician we get that
$$
E\left[\frac{1}{X+1}\right]=\sum_{k=0}^n\frac{1}{1+k}\binom{n}{k}p^k(1-p)^{n-k}
$$
which should be computed. To do that, try and write $\frac{1}{k+1}$ as part of the term $\binom{n}{k}$ by noting that
$$
\frac{1}{k+1}\binom{n}{k}=\binom{n+1}{k+1}\frac{1}{n+1}.
$$
A: we know :$\frac{1}{X+1}=\int_{0}^{1}s^{X}ds$
$E(\frac{1}{X+1})=\int_{0}^{1}E(s^{X})ds$
$MGF=E(e^{tx})=(q+pe^t)^n;(p+q=1)$
Also
$E(s^{X})=(q+ps)^n$
Thus
$E(\frac{1}{X+1})=\int_{0}^{1}(q+ps)^nds=\dfrac{(q+ps)^{n+1}}{p(n+1)}\bigg]_{0}^{1}=\dfrac{(q+p)^{n+1}}{p(n+1)}-\dfrac{q^{n+1}}{p(n+1)}$
$=\dfrac{1}{p(n+1)}-\dfrac{q^{n+1}}{p(n+1)}$
A: By definition of Expectation, $\mathbb{E}(\frac{1}{1+X})$ should look like $\sum \frac{1}{1+x}\cdot p(x)$. In fact,
$$\mathbb{E}(\frac{1}{1+X})=\sum_{k=0}^{n}\frac{1}{1+k}\cdot {n \choose k}p^k(1-p)^{n-k}=\frac{1}{(n+1)p}\cdot \sum_{k=0}^{n}{n+1 \choose k+1}\cdot p^{k+1}(1-p)^{n-k}=\frac{1}{(n+1)p}\cdot(1-(1-p)^{n+1}) $$
A: Law of the unconscious statistician: if $X$ has probability mass function $p(x)$, then
$E[f(X)] = \sum_x p(x) f(x)$.
