Let $X_1$ and $X_2$ be independent normal random variables with
$EX_1 = EX_2 = 0$
$Var(X_1) =\sigma_1^2$ and $Var(X_2) =\sigma_2^2$
Let $Y_1 = X_1 + X_2, Y_2 = X_1 - X_2$
Find the joint distribution of $Y_1$ and $Y2$
I know that the sum ($Y_1$) and the difference ($Y_2$) are both normally distributed ~ $N(0, \sigma_1^2+\sigma_2^2)$
However, I am not sure if $Y_1$ and $Y_2$ are independent and how their joint distribution would be if they are not independent. I'd appreciate if anyone can help me.