The result works for $(X_1,X_2)$ jointly normal (i.e. with identical marginscorrelation, $-1<\rho<1$), with common $\sigma$.
If you know a couple of basic results, this is about all you need:
$\quad\quad\quad$ 
dobiwan's approach is essentially fine - it's just that the result is more general than the case dealt with there.
The result works for $(X_1,X_2)$ jointly normal with identical margins.
If you know a couple of basic results, this is about all you need:
$\quad\quad\quad$
The result works for $(X_1,X_2)$ jointly normal (i.e. with correlation, $-1<\rho<1$), with common $\sigma$.
If you know a couple of basic results, this is about all you need:
$\quad\quad\quad$
dobiwan's approach is essentially fine - it's just that the result is more general than the case dealt with there.
