= Raw moment recursion for a compound Poisson distribution
{title2=$m_k=\lambda\sum_{j=0}^{k-1}\binom{k-1}{j}\mu_{j+1}m_{k-1-j}$}
Write $m_k=\mathbb E S^k$, $m_0=1$, and $\mu_j=\mathbb E X^j$. Apply the <Poisson size-bias identity> with $h(s)=s^{k-1}$, and expand $(S+X)^{k-1}$ using the <binomial theorem> and <independence>. A finite $k$th severity <moment> suffices. In particular, the aggregate <expected value>, <variance>, and third <central moment> are $\lambda\mu_1$, $\lambda\mu_2$, and $\lambda\mu_3$, respectively; the latter two use raw severity <moments>.
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