Consider three Gaussian variables $X_1,X_2,X_3$ with $\mathbb{E}[X_i]=0$ and $\mathbb{E}[X_iX_j]=\rho_{ij}$ for $i,j=1,2,3$. Then, three new variables are defined: $$ \left\{ \begin{array}{l1} Y_1 =X_1 \\ Y_2= X_1+X_2\\ Y_3=X_2+X_3 \end{array} \right.$$ Determine whether $Y_i$ with $i=1,2,3$ are jointly Gaussian and find $f_{Y_1Y_2Y_3}(y_1,y_2,y_3)$.
I don't know where to start. $X_1$,$X_2$ and $X_3$ are not independent as they are correlated (it is not affirmed that $\rho_{ij}=0$ for any pair of values $i$ and $j$), so I can't just multiply their marginal density functions. Apart from that, I don't know if they are jointly Gaussian because they are not independent, so I'm pretty stuck here.
Is it possible to solve this without assuming that $X_1$,$X_2$ and $X_3$ are jointly Gaussian?