I was hoping someone could propose an argument explaining why the random variables Y1=X2−X1 and Y2=X1+X2, Xi having the standard normal distribution, are statistically independent. The proof for that fact follows easily from the MGF technique, yet I find it extremely counter-intuitive.
I would appreciate therefore the intuition here, if any.
Thank you in advance.
EDIT: The subscripts do not indicate Order Statistics but IID observations from the standard normal distrubution.
This is standard normal distributed data:
Notice that the distribution is circulary symmetric.
When you switch to Y1=X2−X1 and Y2=X1+X2, you effectively rotate and scale the axis, like this:
This new coordinate system has the same origin as the original one, and the axis are orthogonal. Due to the circulary symmetry, the variables are still independent in the new coordinate system.