Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 218 / 2 / a / Solution

For user $j$ and offer $r$, both fits useThe first model treats each click as Bernoulli$(p_j)$. The grouped model treats the click count $C_j=m_j{\mathrm{prop}}_j$ as $\mathrm{Bin}(m_j,p_j)$. Their likelihoods differ only by binomial coefficients independent of the parameters, so their fitted coefficients agree.