= Compound binomial distribution
{title2=$M_S(t)=(1-p+pM_X(t))^n$}
= Binomial compound sum
{synonym}
For an independent <binomial distribution> count $N\sim\operatorname{Binomial}(n,p)$, the aggregate can also be represented as $\sum_{j=1}^n B_jX_j$, with independent parameter-$p$ <Bernoulli random variables> $B_j$ and independent claim sizes. Its <cumulant-generating function> is $n\log(1-p+pM_X(t))$. Its third <cumulant> is $n(pm_3-3p^2m_1m_2+2p^3m_1^3)$ in severity <moments>. Constant positive claims $d$ give $np(1-p)(1-2p)d^3$, which is negative when $p>1/2$. Thus positive claim sizes do not by themselves force a positive third <central moment>.
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