So I have this problem that I am doing for practice, and it asks me to find the mgf (moment generating function) of $X_1$ and then it asks me to find the mgd of $(X_1,X_2)$ so what I am wondering is the following.
The joint pdf is: $f_{X_1,X_2}(x_1,x_2) = exp(-x_1-x_2) $ for $x_1>0$ and $x_2>0$ so what I am wondering is that since $x1$ and $x2$ are independent, would I use $g(x_1) = e^{-x_1}$ in the mgf formula to find the mgf of $X_1$ and then for the mgf of $(X_1,X_2)$ would I use the pdf in the mgf formulas ? If this is incorrect what is wrong with the way I am thinking about this problem ? Any help would be greatly appreciated.