Let \(\{X_r : r=1,2,3,\cdots\}\) be a collection of i.i.d. random variables. Let \(G(s))\) be the generating function  of \(X_1\) ( i.e. \(G(s)=\mathbf{E} (s^{X_1})\) ), and hence; each of the \(X_r\)’s. Let \(N\) be an additional random variable taking values in the non-negative integers which is independent of all of the \(X_r\). Let \(H(s)\) be generating function of \(N\).

1. Define the random variable \[ T=\sum_{k=1}^N X_k\] where \(T=0\) of \(N=0\).  For any fixed \(s>0\), calculate \( \mathbf{E}[ s^T | N]\). Show that the generating function of \(T\) is  \(H(G(s)) \).
2. Assume that each claim that a given insurance company  pays is independent and distributed as an exponential random variable with parameter \(\lambda\). Let  the number of claims  in a given  year be distributed as geometric  random variable with parameter \(p\). What is the moment generating function of the total amount of money payed out in a given year ? Use your answer to identify the distribution of the total money payed out in a given year.
3. Looking back at the previous part of the question, contrast your answer with the result of adding a non random number of exponential together.