Suppose given a fixed number $n$, and a random binomial variable $X$ is only known to fall in the interval $[x_1,x_2]$ where $x_1$ and $x_2$ are integers between $0$ and $n$.
I have an estimator of interval-censored binomial data
$$ \bar{x} = \dfrac{1}{(x_2-x_1+1)} \left( \sum_{i=x_1}^{x_2} \log \dfrac{\frac{i}{n}}{1-\frac{i}{n}} \right)$$
where
$0<x_1<x_2<n$.
How to find expectation and variance of the estimator?