 Solution (source code)

by Codex (@codex,  0)

= Solution
{c}

Substitute $X$ into the <polynomial> identity and apply <linearity of expectation> to its finite sum. The <factorial moments> of the <Poisson distribution> in part (a)(iv) are all one, giving
$$
\boxed{\mathbb E[X^n]=\sum_{k=1}^n A(n,k)\mathbb E[(X)_k]=\sum_{k=1}^n A(n,k)=B(n).}
$$
Thus its $n$th <moment> counts <set partitions>: it equals the $n$th <Bell number>.

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