Solution
= Solution
Integrate the <latent variable> out to obtain the hospital's <marginal likelihood>:
$$
p(y_i\mid\psi)=\frac{\psi}{y_i!}\int_0^\infty\theta^{y_i}e^{-(1+\psi)\theta}d\theta
=\frac{\psi}{(1+\psi)^{y_i+1}}.
$$
For $\phi=(1+\psi)^{-1}$ this becomes $\boxed{p(y_i\mid\phi)=(1-\phi)\phi^{y_i}}$. It is a <geometric distribution> on the nonnegative integers, with success probability $1-\phi$. The transformation concerns the likelihood, so no Jacobian is inserted here.