Let $X_{1} ..., X_{n}$ be a sample from $U([0,\theta])$ for $\theta>0$ Find an estimator of $\theta$ by using the method of moment and next compute probability that $\theta_{0} < X_{n:n}$ for $n>2$ ,where $\theta_{0}$ is an estimator.
First party is simple and we have that $\frac{\theta_{0}}{2}$ is an average of $X$.Probably, the second one as well, however I have no idea, what is going on there.
$X_1, \dots, X_n$ an iid sample from $U(0, \theta)$. Then we have $E X_i = \theta/2$ so that $E \bar{X} = \theta/2$ where $\bar{X}$ is the arithmetic mean, so the MME is $\hat{\theta}_{\text{MME}}=2\bar{X}$.
Then, if $\theta_0$ denotes that MME (and $X_{n:n}$ denotes the maximum observation), the second question asks for $$ P(\max(x_1, \dots, X_n) > 2\bar{X}) $$ which I will leave for others as an exercise ...
