Calculate expectation of a function of binomial random variables

Question

Let be $X_1,...,X_n$ n independent random variables from a bionomial distribution $Bin(10,\theta)$, how can I calculate or approximate the following expectation: $$ E\left[{1\over{1+\overline{X}_n}}\right]? $$

Answer 1

Sounds an application of the delta method:

https://en.wikipedia.org/wiki/Delta_method#:~:text=In%20statistics%2C%20the%20delta%20method,limiting%20variance%20of%20that%20estimator.

By CLT:

$\sqrt{n}(\overline{X}-10\theta) \rightarrow_D N(0,10\theta(1-\theta))$

and calling $g(x)=1/(1+x)$

$\sqrt{n}(g(\overline{X})-g(10\theta)) \rightarrow_D N(0,10\theta(1-\theta)g'^2(10\theta))$

That is:

$\sqrt{n}\left(\frac{1}{1+\overline{X}}-\frac{1}{1+10\theta}\right) \rightarrow_D N\left(0,\frac{10\theta(1-\theta)}{(1+10\theta)^4}\right)$