I'm going through an old exam in probability theory. The question reads:
Let $X_1,\,X_2,\,X_3$ be $N(1,\,1)$ IIDs. Determine the characteristic function of the vector $U = X_1 + X_2 + X_3,\, V = X_2 + X_3,\,W = X_3$.
I tried using the transformation theorem to get the density of the vector $(U,V,W)$ and proceeding from there, but couldn't get it to work. I'm sure there's a really short solution to this problem since it's, judging by the number of points it gives, considered an easy exercise.