Since $X_j\sim\operatorname{Poisson}(n_j\lambda_j)$, the rate MLE has asymptotic variance $\lambda_j/n_j$. The <delta method> for the logarithm gives
$$
\operatorname{Var}(\log\widehat\lambda_j)\simeq\frac1{n_j\lambda_j}.
$$
Independence of the arms therefore yields
$$
Z=\frac{\log\widehat\lambda_T-\log\widehat\lambda_C}
{\sqrt{1/(n_T\widehat\lambda_T)+1/(n_C\widehat\lambda_C)}}.
$$
Under $H_0:\lambda_T=\lambda_C=\lambda$, the asymptotic variance of the log rate ratio is
$$
\frac1{n_T\lambda}+\frac1{n_C\lambda}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 207 / 3 / d / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-207/3/d/solution