$X_1, X_2$ are independent standard normals. Define $$W_1 = X_1^2 + X_2^2$$ and $$W_2 = (X_1^2 - X_2^2)/(X_1^2 + X_2^2)$$
How can I show that $W_1$ and $W_2$ are independent using the gamma distribution?
So far we know that $X_1^2, X_2^2 $ are both distributed like ~ $2*Gamma(.5)$ due to the relationship between Gamma to Chi-Sq.
$ie: X_1^2$ ~ $G_1$ and $ X_2^2$ ~ $G_2$
This gives us: $W_1 = G_1 + G_2$ and $W_2 = (G_1 - G_2)/(G_1 + G_2$. Where can I go from here?