Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 13J / a / Solution

A generalized linear model assumes independent responses from one exponential dispersion family,where ${\sigma }_i^2=a_i{\sigma }^2$, ${\mu }_i=\mathbb{E}[Y_i]=b^{\prime }({\theta }_i)$, and a link function relates the mean to the linear predictor by $g(\mu )=X\beta$. It follows from normalization of the density after shifting the natural parameter thatIndependence therefore gives the joint moment-generating functionwhenever every shifted natural parameter lies in its natural-parameter domain.