What is meant by an exponential dispersion family? Show that the family of Poisson distributions with parameter $\lambda$ is an exponential dispersion family by explicitly identifying the terms in the definition.

Find the corresponding variance function and deduce directly from your calculations expressions for $\mathbb{E}(Y)$ and $\operatorname{Var}(Y)$ when $Y \sim \operatorname{Pois}(\lambda)$.

What is the canonical link function in this case?