 Solution

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.