Solution
= Solution
Part iv gives
$$
\mathbb E[v_+\mid\theta_0,\phi_0]
=n\frac{\theta_0}{\theta_0+\phi_0}
\left(1-e^{-(\theta_0+\phi_0)c}\right).
$$
Therefore the information requirement is
$$
n\frac{1-e^{-(\theta_0+\phi_0)c}}
{\theta_0(\theta_0+\phi_0)}
\geq I_0.
$$
The required integer sample size is
$$
n=
\left\lceil
\frac{I_0\theta_0(\theta_0+\phi_0)}
{1-e^{-(\theta_0+\phi_0)c}}
\right\rceil.
$$