Suppose X = (X1. ...; Xn) is a random sample from the distribution f(x; 0) = (x -1)03(1-0)3-2, x=2,3....;0

Suppose X = (XL, , Xn) is a random sample from the distribution (x - -or-2 .c 23, The prior distribution for O is a beta distribution: r(2n + 5) 93 (1 9)2n OSOS 1 and the loss from estimating O by is (9 )2. a) b) c) d) Find the posterior distribution of 9 conditional on X. Hence, calculate the Bayes estimator 9 of 9. If the observations were 2, 2, 3, 2, 4, 2, 3 find the estimate of O for this data. It was claimed that 9 is less than 0.4. Test the claim via Bayesian testing with a zero-one loss, using the data from c). Hint: You may find it helpful to use the R function pbeta in your answer.