General formulation of the product arbitrary moment generating functions X and Y

Question

Given that X and Y are independent random variables, with moment generating functions M and N, respectively, is there a generic formula to produce the moment generating function of X*Y?

Answer 1

You could use the law of total expectation. For instance, if $Y$ is a discrete r.v \begin{align} M_{XY}(t) = \mathbb{E}[e^{tXY}] &= \sum_{y} \mathbb{E}[e^{tXY}|Y=y]\mathbb{P}[Y=y] \\ &=\sum_{y} \mathbb{E}[e^{tXy}]\mathbb{P}[Y=y] \\ &=\sum_{y} M_{X}(ty) \mathbb{P}[Y=y], \end{align} where the second equality follows from the independency of $X$ and $Y$. Similarly, for a continuous r.v $Y$ with density $f_Y$, $$M_{XY}(t) = \int_{y} M_X(ty)f_Y(y)dy.$$