To study the mean of a population variable $X$, $\mu = E(X)$, a simple random sample of size $n$ is considered. Imagine that we do not trust the first and the last data, so we consider the following statistics:
$\sim X$ = $\frac{1}{n-2} \sum_{j=2}^{n-1} Xj$ = $\frac{X_2+....X_{n-1}}{n-2}$
Calculate the expectation and the variance of this statistic. Calculate the mean square error (MSE) and its limit when $n$ tends to infinite. Study the consistency. Compare the previous error with that of the ordinary sample mean.
I calculated the expectation of this statistic doing : $E(\sim X)$ = $\frac{1}{n-2} \sum_{j=2}^{n-1} Xj$ = $\frac{(n-2)\mu}{n-2} = \mu$
I do not know how to find out the variance of this statistics and consequently how to continue the exercise.
Any help, suggestion would be appreciate