Let $X_1, X_2,…$ be independent, identically distributed random variables and let $N$ be a random variable which takes values in the positive integers and is independent of the $X_i$. Find the moment generating function of $S = X_1 + X_2 + · · · + X_N$ in terms of the moment generating functions of $N$ and $X_1$, when these exist.
The hint in the book says: "This is basically the same argument as in the random sum formula, $M_S(t)=G_N(M_X(t))$."
I know that the mgf is defined as $M_X(t)=\mathbb{E}(e^{tX})$. I'm not sure what this question is asking me to do. Do I just write $M_S(t)=G_N(M_X(t))$ and solve for the RHS? The I think N is a Poisson random variable since it take any values {1, 2, …}. In which case the outer part $G_N(s)$ equals $e^{\lambda (s-1)}$. I'm not sure about $M_X(t)$.
Best Answer
Let the moment generating function of S is $M_S(t)$ then $$ M_S(t)=\mathbb{E}\left(\exp{\left(t\sum_{i=1}^N X_i\right)}\right) = \mathbb{E}\left(\mathbb{E}\left(\exp{\left(t\sum_{i=1}^N X_i\right)}\big|N\right)\right) = \mathbb{E}\left(\prod_{i=1}^N\mathbb{E}\left(\exp{\left(t X_i\right)}\big|N\right)\right) = \mathbb{E}\left(\prod_{i=1}^NM_{X}(t)\right)= \mathbb{E}\left(\left(M_{X}(t)\right)^N\right)= G_N(M_{X}(t)). $$ The second inequality is due to the tower property of expectation. The third inequality is due to the fact that $X_i$'s are independent of each other and also independent of $N$. The fourth inequality is again due to the fact that $X_i$'s are independent of $N$. $M_X$ is the moment generating function of $X_i$'s and $G_N$ is the probability generating function of $N$.