Haven't used this for a while.. not sure if i get the texing.

I prove first that (this was fine)

$\displaystyle \frac{1}{2n(n-1)} \sum^n_{i=1}\sum^n_{j=1}(X_i-X_j)^2 = S^2$

Then i am to calculate, assuming all the $\displaystyle X_i's $ have finite fourth moment, denoting $\displaystyle \theta_1 =EX_i, \theta_j=E(X_i - \theta_1)^j, j=1,2,3$, the

$\displaystyle Var(S^2)=\frac{1}{n}\left(\theta_4-\frac{n-3}{n-1}\theta_2^2 \right)$. I first used $\displaystyle Var(S^2)=E(S^4)-E(S^2)^2$ and tried to calculate $\displaystyle E(S^2)$ but

I don't really get how these cross terms work. My first attempt i got$\displaystyle \theta_2/(n-1)$ but i'm only assuming that's wrong.