the variance of a function

Question

I don't understuand how we can get get this variance :

if $f : R \rightarrow R$ and $X_1,...X_n$ i.i.d then $$ \operatorname{var}\left(f\left(X_{1}\right)\right)=\frac{1}{2} \mathbb{E}\left[\left(f\left(X_{1}\right)-f\left(X_{2}\right)\right)^{2}\right] $$

Answer 1

Your statement is a special case of the following general result:

If $U$ and $V$ are i.i.d. then $\text{Var}(U) = \frac{1}{2} E[(U-V)^2]$.

Just expand the right-hand side to obtain $\frac{1}{2}(E[U^2] - 2E[UV] + E[V^2])$, and note that $E[UV] = E[U]E[V]=(E[U])^2$ and that $E[V^2]=E[U^2]$.