According to this answer,
$X+Y$ and $X−Y$ are Gaussian random variables, so that $(X+Y)^2$ and $(X−Y)^2$ are Chi-square distributed with 1 degree of freedom, where $X\sim N(a,b)$ and $Y\sim N(c,d)$
I'm confused since while I understand that $(X+Y)$~Normal$(a + c, b + d)$ and $(X-Y)$~Normal$(a + c, b + d)$, I do not understand why $(X+Y)^2$ and $(X-Y)^2$ are chi-squared random variables.
As far as I know, only squares of standard normal random variables can be chi-squared distributed, and $(X+Y)^2$ and $(X-Y)^2$ are, in general, not standard normal random variables.
Am I misunderstanding / missing something here?
