Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 219 / 2 / b

First integrate out ${\eta }_i$: conditional on ${\xi }_i$, $y_i\sim N(\alpha +\beta {\xi }_i,s^2)$ with $s^2={\sigma }^2+{\sigma }_y^2$. Since $(x_i,y_i)$ is an affine transformation of independent normal variables, it is bivariate normal with meanand covarianceIts determinant simplifies toFor $r_i=(x_i-\mu ,y_i-\alpha -\beta \mu {)}^T$, the observed-data likelihood is therefore