Hi,

I have an iid sample from $\displaystyle U(\theta_1, \theta_2) $ and found the MM of $\displaystyle (\theta_1, \theta_2) $ to be:

$\displaystyle

\theta_1 = \bar X_n - \sqrt {3 Sn^2} $ and $\displaystyle \theta_2 = \bar X_n + \sqrt {3 Sn^2}

$

In our solutions, we are told to notice that

(i) $\displaystyle \frac {1} {n} \sum_i {(X_i)^2} - (\frac {1} {n} \sum_i {X_i})^2 = \frac {n-1} {n} {S_n}^2

$

and

(ii) $\displaystyle E(3(\frac {1} {n} \sum_i {(X_i)^2} - (\frac {1} {n} \sum_i {X_i})^2)) = \frac {n-1} {n} \frac {(\theta_2 - \theta_1)^2} {4}

$

I'd love to have help on how to actually see this! Thank you!