Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 4 / 17H / b / Solution

For known ${\lambda }_0$, the likelihood ratio of $\pi =1/2$ to $\pi =0$ iswhich increases with $n_0$. Reject for $n_0>c$, with randomization at $n_0=c$ chosen so that the null rejection probability is exactly $\alpha$. Since the zero indicators form a Bernoulli family with success probability increasing in $\pi$ and have a monotone likelihood ratio in $n_0$, the Karlin-Rubin theorem makes this test uniformly most powerful for $\pi >0$.