[This problem is about (1) UMP hypothesis tests and (2) confidence intervals obtained by inversion. And a Bayes credible interval (set).] Let X1,...,Xn be a random sample from the density f(x|✓) = ✓2 xe✓xI(0,1)(x), where ⌦ = {✓ : ✓ > 0} is the parameter space. a) Consider the problem of testing H0 : ✓ = ✓0 versus H1 : ✓ 6= ✓0. Derive the size ↵ likelihood ratio test of these hypotheses. Show that the rejection region can be written in terms of the statistic Pn i=1 Xi. Give an explicit form for the rejection region in terms of percentiles of a 2 distribution. Hint: The random variable 2✓ Pn i=1 Xi has a chi-square distribution with 4n degrees of freedom. b) Now consider testing H0 : ✓  ✓0 versus H1 : ✓ > ✓0. Doe