Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 29L / c / i / Solution

For $y=0,1,\dots$,so $Y_i$ is geometric with success probability $1-e^{-\theta }$. The log likelihood isIts score equation givesand henceWhen $\overline{Y}=0$, the extended MLE is $+\infty$; this event has probability tending to zero.