= Solution
An <exponential dispersion family> has density or mass function
$$
p(y;\theta,\phi)=\exp\left\{\frac{y\theta-b(\theta)}{a(\phi)}+c(y,\phi)\right\},
$$
with support independent of the natural parameter $\theta$. Usually $a(\phi)=\phi/w$ for a known weight $w$. For the <Poisson distribution>, identify
$$
\theta=\log\lambda,\qquad b(\theta)=e^\theta,\qquad a(\phi)=1,\qquad c(y,\phi)=-\log(y!),\quad y\in\mathbb N_0.
$$
The dispersion is fixed at one. Differentiating the normalizing identity gives $\mathbb EY=b'(\theta)$; differentiating again gives $\operatorname{Var}Y=a(\phi)b''(\theta)$. Here both <derivatives> equal $e^\theta=\lambda$, so
$$
\boxed{\mathbb EY=\lambda,\qquad\operatorname{Var}Y=\lambda,\qquad V(\mu)=\mu.}
$$
The <variance function> expresses $b''(\theta)$ in terms of the mean $\mu$. The <canonical link function> maps the mean to the natural parameter, so \b[the Poisson canonical link is $g(\mu)=\log\mu$].
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