Hi,

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

<br />
\theta_1 = \bar X_n - \sqrt {3 Sn^2} and  \theta_2 = \bar X_n + \sqrt {3 Sn^2} <br />

In our solutions, we are told to notice that

(i)  \frac {1} {n} \sum_i {(X_i)^2} - (\frac {1} {n} \sum_i {X_i})^2 = \frac {n-1} {n} {S_n}^2<br />

and

(ii)  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}<br />

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