4. (Total of 20 Marks) Assume our data Y given X is distributed Y|X = x ~ Binomial(n, p = x) and we chose the prior to be X ~ Beta(o, B). Then the PMF for our data is Prix(yz) = (" (1 - x)"-y, for re [0, 1], y e {0, 1, ..., n). and the PDF of the prior is given by _[(a + B) fx (I ) = T( @ ) I ( B ) "-1(1 - x)6-1, for 0 0, 3 >0. Note that, E[X] = 24, and Var(X ) = (at8)-(a+8+1)' aB (a) Show that the posterior distribution is Beta(a + y, B +n - y). (b) Write out the PDF for the posterior distribution fxy(xly). (c) Find mean and variance of the posterior distribution, E[X|Y = y] and Var(X|Y = y ). (d) Show that the posterior mean, E[X|Y = y], is a weighted sum of the data mean, y = y, and the prior mean, where the sum of the weights is equal to 1