Consider the collective-within-individual risk model $S = S_1 + .... + S_n$, where $S_i$ has a compound distribution with $N_i$ distributed Geometric($p$) and $X_i$ is distributed Exponential($m$) for all $i = 1,....,n$.
Show that $S$ has a compound distribution with $N$ distributed Binomial($n$, $1-p$) and X distributed Exponential($pm$).
I'm not sure how to do this. I tried using the mgf technique to get $N$ but I end up with a negative Binomial not a Binomial.