Suppose that we have n i. i.d. random variables x1, x2, xn, each with the following probability function (as usual, 0 < θ < 1 is unknown): 2 (a) if the prior distribution (pd. f) of θ ls uniform(q 1), what is the posterior pd. f fex(le)? hint: consider a generic set of sample values x - ( , tn) and let ni be the number of the observations in the sample that are equal to i (with i -0,1,2,3). now apply bayes' rule fx(x) "ignoring" the denominator, which is part of the normalizing constant of the posterior density. by inserting the f(x|θ)'s, factoring out the parts that do not depend on θ and merging then into the normalizing constant, you should recognize that the posterior is a beta(a, b) density. what are o and b? (b) if you want a number rather than a p. d.f. as your estimate, there are several options: the conditional expectation is one. compute θ e'(9x-z), and express it in terms of no, , n3. now write the conditional expectation (9 x) as a r. v. and show that, for large n, it is approximately equal to al. (c) finally, assume that we have a sample size of n 10 with sample values æ (3.0, 2, 1,3,2, 1,0,2, 1) compute the expression for the posterior density and roughly plot it. also, compute the mode of the posterior, i. e. the value of θ that maximizes the posterior pdf. felx(81x)