Define the concepts of an exponential dispersion family and the corresponding variance function. Show that the family of Poisson distributions with parameter $\lambda>0$ is an exponential dispersion family. Find the corresponding variance function and deduce from it expressions for $E(Y)$ and $\operatorname{Var}(Y)$ when $Y \sim \operatorname{Pois}(\lambda)$. What is the canonical link function in this case?