Independence of Transformed Bivariate Normal

by Carol Eisen   Last Updated November 14, 2017 18:19 PM

Let $\alpha$ and $\beta$ be jointly distributed by the bivariate normal distribution.

Let A and B be jointly distributed by the standard bivariate normal distribution, where

$A=\frac{\alpha-\mu_\alpha}{\sigma_\alpha}$ and $B=\frac{\beta-\mu_\beta}{\sigma_\beta}$

Is is sufficient to show that A+B and A-B are independent, given $\sigma_A=\sigma_B$, in order to establish that $\alpha+\beta$ and $\alpha-\beta$ are independent if $\sigma_\alpha=\sigma_\beta$?

My intuition is that it should be sufficient as the transformation from A to $\alpha$ and B to $\beta$ are independent of each other. However, I can't find any rigorous proof / theory about this.

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