2. Suppose that X1, .... Xn are a random sample of size n with a Beta(#, 1) pdf and Yl, . ... Ym are an i.i.d sample of size m with a Beta(0, 1) density. Also assume that the X's are independent of the Y's. Consider the testing of Ho : p = 0 vs H1 : p / 0. Hint: If X ~ Beta(o, 8) then its density function is 1 f(r; a, B) = B(a, B) where 0 0, 8 > 0 and B(a, 8) = [to-1(1 -{)6-1 is the Beta function. a) Calculate the MLE of 0 and a and denote them by 0 and f. Then calculate the MLE under the restriction of the null hypothesis Ho and denote it by do. b) Find the LRT statistic A(X, Y). You may keep your final answer in terms of f, 0 and en. c) Show that the test can be based on the statistic T = El- In Xi EL, In Xi+ En, In Ya' with rejection region of the form T