Assume $Z_1$, $Z_2$ and $Z_3$ are independent standard normally distributed $N(0,1)$. We have $X=Z_1+Z_2$ and $Y=Z_1+Z_3$. Need to show $X$ and $Y$ are bivariate normally distributed with $((0,0)', \Sigma=\begin{bmatrix} 2 & 1 \\ 1&2 \end{bmatrix})$
Always work with problems with two $Z$ variables. Any suggestion?