I need to find the joint distribution of $X$ and $Y$ where $X=Z_1+Z_2$ and $Y=Z_1-Z_2$. And $Z_1,Z_2$ are indepent and standard normally distributed.
I've proved that $X$ and $Y$ are both normally distributed with mean $0$ and $\sigma=\sqrt{2}$. I've also proved that the covariance is $0$, but I am not sure how that helps. I don't require a proof just some ideas that I might be missing so I can continue.
Thank you!
Hint: $E e^{tX+sY}=Ee^{(t+s)Z_1+(t-s)Z_2)}=Ee^{(t+s)Z_1}Ee^{(t-s)Z_2)}=e^{(t+s)^{2}/2}e^{(t-s)^{2}/2}$. Simplify this to get $e^{t^{2}}e^{s^{2}}$ and conclude that $X,Y$ are actually independent and that they have a joint normal distribution.