The pf for the Poisson distribution is
$$
p_{k}=\frac{e^{-\lambda} \lambda^{k}}{k !}, \quad k=0,1,2, \ldots
$$
The probability generating function from Example $3.6$ is
$$
P(z)=e^{\lambda(z-1)}, \quad \lambda>0 .
$$
The mean and variance can be computed from the probability generating function as follows:
$$
\begin{aligned}
\mathrm{E}(N) &=P^{\prime}(1)=\lambda \
\mathrm{E}[N(N-1)] &=P^{\prime \prime}(1)=\lambda^{2} \
\operatorname{Var}(N) &=\mathrm{E}[N(N-1)]+\mathrm{E}(N)-[\mathrm{E}(N)]^{2} \
&=\lambda^{2}+\lambda-\lambda^{2} \
&=\lambda
\end{aligned}
$$
4CCM141A COURSE NOTES ：
It is not difficult to show that the probability generating function for the negative binomial distribution is
$$
P(z)=[1-\beta(z-1)]^{-r} .
$$
From this it follows that the mean and variance of the negative binomial distribution are
$$
\mathrm{E}(N)=r \beta \text { and } \operatorname{Var}(N)=r \beta(1+\beta)
$$