I'm seeing in general that for moment generating functions, the mgf of $X+Y$ where $X,Y$ are independent random variables is $M_{X+Y}(t) = M_X(t)M_Y(t)$. I'm also seeing that the joint mgf is given by $M_{(X,Y)}(t) = M_X(t_1)M_Y(t_2)$.
I'm not understanding why these two things would have the same formula. That is, why does $M_{(X,Y)}(t) = M_{X+Y}(t)$ for independent random variables? Would appreciate both a mathematical and heuristic explanation of why these are the same. I believe I may be making an error, however, in thinking the two formulas are the same.
Best Answer
They don't have the same formula. $M_X(t) M_Y(t)$ is a function of a single variable $t$, whereas $M_X(t_1) M_Y(t_2)$ is a function of two variables, $t_1$ and $t_2$. You might say, but what if $t_1 = t_2$? Then yes, you get the MGF of $X + Y$. But they are no more the same as if you were to claim that $f(x,y) = x^2 + y^2$ is equivalent to $g(x) = 2x^2$. They are drastically different functions.