Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 7 / a

For the exponential distribution with rate $\theta$, the change of variables $u=\theta x$ givesThe integrals are one and two by integration by parts. Hence $\boxed{\mathbb{E}[X]=1/\theta ,\mathrm{Var}(X)=1/{\theta }^2}$.

For an estimator $T$ of $\theta$, its bias and mean squared error areWhen the second moment exists, expanding around ${\mathbb{E}}_{\theta }[T]$ gives the bias-variance decomposition ${\mathrm{MSE}}_{\theta }(T)={\mathrm{Var}}_{\theta }(T)+b_{\theta }(T{)}^2$.

For the independent random sample, the likelihood function on its support is $L(\theta )={\theta }^n{e}^{-\theta Y}$, where $Y=\sum X_i$. Its log-likelihood is $\ell (\theta )=n\log \theta -\theta Y$, withAlmost surely $Y>0$, and the unique stationary point is consequently the unique global maximum:The exceptional all-zero sample has probability zero; if $Y=0$ were nevertheless supplied, there would be no finite maximizing rate.