Q:

Proof involving the difference of iid normal random variables

I am trying to prove that, for two i.i.d.  random variables $X_1,X_2 \sim N(\mu, \sigma^2)$, their difference divided by $\sqrt  2$ is standard normal, i.e. $\frac {X_1-X_2}{\sqrt 2} \sim N(0,1)$.
I am trying to show it using convolution, but am getting caught up in the algebra of trying to get it into a recognizable form.  I think there is a more simple way using the definition of expectation, but when I try it, it doesn't quite come out right as I get $N(0, \sigma^2)$ instead of $N(0,1)$.

A:

$Ee^{itX_1}=e^{it\mu}e^{-t^{2}\sigma^{2}/2}$. Hence $Ee^{it(X_1-X_2)/\sqrt2 \sigma)}=Ee^{itX_1/\sqrt2 \sigma)}Ee^{-itX_2/\sqrt2 \sigma)}=(e^{-t^{2}/4})^{2}=e^{-t^{2}/2}$ which is the characteristic function of $N(0,1)$.

