How do you find variance? Let $\{X_n\}$ be a sequence of independent identically distributed random variables with mean $\mu$ and finite variance.

$$T_n= \binom{n}{2}^{-1} \sum_{1\leq i&lt; j\leq n} {X_{i}X_{j}} $$


  I have to calculate $Var(T_{n})$.
$$Var(X_1 X_2)=E(X_1^2 X_2^2)-E(X_1X_2)^2=(\sigma^2+\mu^2)^2-\mu^4=\sigma^4+2\sigma^2\mu^2$$

$$Var(T_n)=\binom{n}{2}^{-1} Var(X_1 X_2)=\frac{\sigma^4+2\sigma^2\mu^2}{n (n-1)/2}$$
$$Var(X_1 X_2)=E(X_1^2 X_2^2)-E(X_1X_2)^2=(\sigma^2+\mu^2)^2-\mu^4=\sigma^4+2\sigma^2\mu^2$$

$$Var(T_n)=\binom{n}{2}^{-1} Var(X_1 X_2)=\frac{\sigma^4+2\sigma^2\mu^2}{n (n-1)/2}$$