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

Using the probability density function of the Erlang distribution, the expected value isHere $u=\theta x$, and two applications of integration by parts give ${\int }_0^{\infty }{u}^2{e}^{-u}du=2{\int }_0^{\infty }u{e}^{-u}du=2$. The boundary terms vanish because exponential decay dominates these powers.