Consider the following:

\[ \begin{aligned} N&\sim\text{Poisson}(\lambda)\\ X_1,\,X_2,\,X_3,\,...&\overset{\text{iid}}{\sim}M\\ S&=\sum\limits_{i=1}^NX_i. \end{aligned} \]

\(N\) is independent of all of the \(X_i\). The \(X_i\) are an infinite sequence of iid random variables each sharing a common moment-generating function \(M(t)=E(e^{tX_1})\). \(S\) is a random sum of random variables. The terms are random, and so is \(N\), the number of terms being summed.

1. What is the MGF of \(S\)?
2. If I had an iid collection of \(S_1\), \(S_2\), …, \(S_m\) each possessing the same distribution that you just derived, what would be the distribution of \(T=S_1+S_2+...+S_m\)?