Solution
= Solution
For the <Poisson distribution> with parameter $1$, the <exponential series> gives
$$
G(s)=\sum_{j=0}^{\infty}\frac{e^{-1}s^j}{j!}
=\boxed{e^{s-1}},\qquad s\in\mathbb R.
$$
Every <derivative> is $G^{(k)}(s)=e^{s-1}$, so the <factorial moments> are
$$
\boxed{\mathbb E[X(X-1)\cdots(X-k+1)]=1\qquad(k\ge0).}
$$
In particular $G'(1)=G''(1)=1$, and the <variance> is
$$
\boxed{\operatorname{var}(X)=1+1-1^2=1.}
$$