For the setting of the previous exercise, show that \(Y\) is distributed according to the mixture with density

\[
m(y)=(1-ho) \phi(y)+ho \psi(y)+o(ho)=\phi(y)(1-ho+ho \zeta(y))+o(ho)
\]

where \(\psi(\cdot)\) is a probability density, \(\zeta(y)=\psi(y) / \phi(y)\) is the density ratio, and \(\zeta(0)=0\). Fill in the details needed to express \(\zeta(\cdot)\) or \(\psi(\cdot)\) as a function of the family \(P_{v}\).