Exponential dispersion family of order one ($p(y;\theta ,\varphi )=\exp \{(y\theta -b(\theta ))/\varphi +c(y,\varphi )\}$)

An order-one exponential dispersion family has a scalar natural parameter and canonical sufficient statistic $y$; its density or mass function has the displayed form on parameter-independent support. Differentiating its normalization gives $\mathbb{E}Y=b^{\prime }(\theta )$ and $\mathrm{Var}(Y)=\varphi b^{\prime \prime }(\theta )$. Known observation weights $w$ replace $\varphi$ by $\varphi /w$. Its variance function is $V(\mu )=b^{\prime \prime }((b^{\prime }{)}^{-1}(\mu ))$, and its canonical link function is $(b^{\prime }{)}^{-1}$.