Solution
= Solution
{c}
For the specified <Poisson distribution>, the <exponential series> gives the <probability generating function>
$$
G(s)=e^{-1}\sum_{j=0}^\infty\frac{s^j}{j!}=\boxed{e^{s-1},\qquad s\in\mathbb R.}
$$
Every <derivative> of $G$ is again $e^{s-1}$, so the <factorial moments> and <variance> are
$$
\boxed{\mathbb E[(X)_k]=1\quad(k\geq1),\qquad \operatorname{var}(X)=1+1-1=1.}
$$
Also $\mathbb E[X]=1$.