Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 44 / 3 / a / Solution

Write the new measurement as $Y_{i1}={\theta }_i+{\epsilon }_{i1}$, with independent ${\theta }_i\sim N(\mu ,{\tau }^2)$ and ${\epsilon }_{i1}\sim N(0,{\sigma }_i^2)$. Adding independent normal distributions gives the prior predictive distribution