Please solve this

Let X1, . . . , X\" be a random sample from the density f(I|3) = 92I_gxf(o,m)($), where Q = {6 : 9 > 0} is the parameter space. a) Consider the problem of testing H9 : 6 = 60 versus H1 : 6 79 60. Derive the size a likelihood ratio test of these hypotheses. Show that the rejection region can be written in terms of the statistic 221:1 Xi. Give an explicit form for the rejection region in terms of percentiles of a x2 distribution. Hint: The random variable 29 E?=1Xg has a chi-square distribution with 411 degrees of freedom. b) Now consider testing H0 : t9 5 90 versus H1 : 9 > 99. Does a uniformly most powerful size 0: test exist in this case? Carefully explain your reasoning, and if your answer is yes, nd the UMP size a test. c) Invert the test in part a] to obtain a (1 a)100%. condence interval for 6. (1) Suppose now that 9 has the following prior density: (9) = 49e-291(0m)(9). Derive the level 1 o: {highest posterior density) Bayes credible region for 6