A random varible X taking values from [0, 1] has Beta distribution of parameters a and B, which we denote by Beta(a, B), if it has PDF fx (x) =- r(a + p) r(a) r (B) x"-1(1 -x)6-1 (2) where I(z) is the Euler Gamma function defined by I(z) = fox-le-* dx. Bob has a coin with unknown probability 0 of heads. Alice has the following Beta prior: J = Beta(1, 2). (3) Suppose that Bob gives Alice the data On = (x1, ..., Xn), which is the outcome of n indepen dent coin flips. Denote Sn = X1 + . . . + Xn. Suppose n = 10 and Sn = 7. (i) Show that Alice's posterior distribution is Beta(8, 5). Namely, OlOn ~ Beta(8, 5). (4) (ii) From the posterior distribution in (i), construct a Bayesian estimator 0 for 0 for which the posterior mean squared error E[(@ -0)2101 (5) is minimized. (You may use the fact that E[Beta(a, B)] = a/ (a + p).) (iii) From the posterior distribution in (i), find the most likely value of 0. Does this agree with the estimated value o in (ii)