Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 44 / 2 / c / Solution

Integrate the latent variable out to obtain the hospital's marginal likelihood:For $\varphi =(1+\psi {)}^{-1}$ this becomes $\boxed{p(y_i\mid \varphi )=(1-\varphi ){\varphi }^{y_i}}$. It is a geometric distribution on the nonnegative integers, with success probability $1-\varphi$. The transformation concerns the likelihood, so no Jacobian is inserted here.