Compound Poisson Distribution

Let \(S_{N}=X_{1}+\dots _+X_{N}\) be the sum of i.i.d. terms where the number of terms \(N\) is random and follows a Poisson distribution \(N\sim \mathcal{P}(\lambda)\). The random variable \(S_{N}\) follows a compound Poisson distribution with PGF given by \[ G_{S_{N}}(t)=e^{\lambda(G_{X}(t)-1)}. \] \begin{proof} The proof of this result follows from past results in our summary of probability generating functions.\end{proof}

For the sum of i.i.d. Bernoulli RVs \(X_{i}\sim \mathcal{B}(1, p)\) the PGF is given by \[ G_{S_{N}}(t)=e^{\lambda((p(t-1)+1)-1)}=e^{\lambda p(t-1)}, \] hence \(S_{N}\sim \mathcal{P}(\lambda p)\).