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(6 points) (Please give your answer to four (4) decimal places where needed) Let X1, ,X,l be iid Bernoulli(p). Then Y = 2L1 X,- is binarniaKn, p). We assume the prior distribution on p is betu(a, [7). As detailed in the lecture notes, the posterior distribution for p, given the sample data y is (ply) ~ beta(y + a,n y + ). For the following questions, suppose that we specify the prior parameters to be a = 1.4 and = 4. and we obtain y = 5 from a sample of size n = 13, (a) (1 MARK) (True or False): The beta(a, [9) prior distribution on p Is a conjugate family for the binomiaKn, p) family. 0 A.True O B. False (b) (1 MARK) Without having seen the data y, what would be our best estimate 01p using only our prior distribution? ' (c) (1 MARK) Using the observed data, what is the maximum likelihood estimate (MLE, trequentist estimate) lor p7 3 (d) (1 MARK) Using the observed data, what is the Bayes estimate (Bayesian posterior estimate) for p? For the following two questions, assume that we have collected a second independent sample of m = 18 iid observations from the same population distribution: X1, ,Xlg iid Bemaullin). For this sample we observe 2.1:, x.' = y = 15 (e) (1 MARK) II we combine all of the data from both samples (total sample size is n+m), what is the MLE of p? (i) (1 MARK) Now treat the posterior distribution for p given the rst observed sample y = 5 as your new 'updated' prior distribution for p, Use this new prior and the second observed dataset with y = 15 to obtain an 'updated' posterior distribution for p. What is the Bayes estimate for p lrorn this 'updated' posterior