Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 30 / 3 / b / Solution

For $y=0,1,2,\dots$ and $\lambda >0$, the Poisson distribution has massMatch this to the exponential dispersion family withThen $\mu =b^{\prime }(\theta )=e^{\theta }=\lambda$ and $V(\mu )=b^{\prime \prime }(\theta )=\mu$. The canonical link function expresses the natural parameter in terms of the mean, so the Poisson canonical link is $g(\mu )=\log \mu$. Its conditional variance equals its conditional mean.