Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 37 / 1 / c / Solution

For the binomial distribution, $G_N(z)=(1-p+pz{)}^n$. Thus the compound binomial distribution hasSet $h(t)=1-p+p{M}_{X_1}(t)$. Since $h(0)=1$ and $h^{(j)}(0)=p{m}_j$, differentiating $n\log h$ three times givesThe first two cumulants, for comparison, are $np{m}_1$ and $n(p{m}_2-p^2{m}_1^2)$.

Choose deterministic claim sizes $X_i=d>0$. Then $S=dN$, so $m_j=d^j$ andThis is strictly negative for $p>1/2$ and $n\ge 1$. A concrete negative-skewness example is $n=1$, $p=3/4$, $d=1$, for whichThere is no contradiction with positive claims: bounded Bernoulli-type totals can have more mass near their upper value and a longer left tail.