Moment Generating Function property for $M_{X-Y}(t)$ the same as $M_{X+Y}(t)$?

Question

I know that if $X,Y$ independent, then the moment generating function ($M_{X+Y}(t)$)

$M_{X+Y}(t) = M_X(t) M_Y(t)$ is true.

Where $M_{X}(t)$ is the moment generating function of $x$ and $M_{Y}(t)$ is the moment generating function of $y$.

But what about the case of

$M_{X-Y}(t) =$ ?

A simple explanation or proof would help greatly! Thanks!

Answer 1

$M_{X-Y}(t) = \frac{M_X(t)}{M_Y(t)}$, under some mild assumptions about zeros. You can prove this by setting $Y = -Z$, where $Z$ is some random variable, and using your observation about $M_{X,Z}$.