A researcher plans to use a random sample X1, ..., Xn of size n from a population with probability density function -0x f(x|0) = { % for x > 0 otherwise, 0 where parameter 0 > 0. - " (a) If the prior distribution for parameter 0 > 0 is (0) = ae 0 > 0 and zero elsewhere, where a > 0, show that the posterior distribution h(0|x1,...,xn) is a gamma distribution with parameters a = n+1 and = a + nx, where x is the sample mean. 4 (b) What is the formula for the Bayes estimator E(0|x1,...,xn)? If n = 4, x = 2 and a = 1, what is the Bayes estimate of 0?