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

Put $S={\sum }_{i=1}^N{x}_i$. Since the observations are independent random variables, their joint probability density function is a product. Therefore the likelihood function and log-likelihood areThese expressions assume positive observed values, as occurs with probability one under this continuous Erlang distribution.