Past exam of the mathematical and Computational Biology course of the University of Cambridge / 2025 / Paper 1 / 2 / c / Solution

The score function and the observed Fisher information areSolving the score equation gives the maximum-likelihood estimateThe positive observed Fisher information means that the log-likelihood is strictly concave on $\theta >0$. Its unique stationary point is therefore the global maximum. Equivalently, the score function changes from positive to negative at this point. At the estimate, $j(\hat{\theta })=S^2/(2N)$.