Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 218 / 1 / b / Solution

Let $x_i^T$ be row $i$ of the design matrix, ${\eta }_i=x_i^T\beta =1/{\mu }_i$, and $\gamma =1/\varphi$. The full log-likelihood isDifferentiation gives the score functionandThe Hessian does not depend on $Y$, so the Fisher information matrix is