## Question

###### (10 points) (10.3.4 from Probability and Statistical Inference) Suppose that Xi, Xn are iid Uniform(0,0) given that 0 where the parameter 0 is unknown. Suppose that has prior distribution Pareto(a , 8) , which has pdf _ given by h(0) Ba' 0-(5+1)1{a 0} where &, 8 are known positive numbers. Recall the sufficient statistic T' X(n) max{X,. X} which is the largest order statistic. (a) Show that the posterior distribution of 0 given x = {21,12, Tn turns out to be Pareto(a' max(t,

(10 points) (10.3.4 from Probability and Statistical Inference) Suppose that Xi, Xn are iid Uniform(0,0) given that 0 where the parameter 0 is unknown. Suppose that has prior distribution Pareto(a , 8) , which has pdf _ given by h(0) Ba' 0-(5+1)1{a 0} where &, 8 are known positive numbers. Recall the sufficient statistic T' X(n) max{X,. X} which is the largest order statistic. (a) Show that the posterior distribution of 0 given x = {21,12, Tn turns out to be Pareto(a' max(t,a), 8' = n + 8), where T = t = max{T1, Infa (b) What is the Bayes estimator for 0? (c) Suppose 10, Xn 1.5, a =4, 8 = 20. What is the posterior probability of Ho 0 < 22

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