Answer Happy

Transcribed Image Text from this Question4. (25 points) Consider three hypotheses about the distribution of X Ho :X fo, Hj: X ~ f1, H2 : X $2, where the definitions of fo, fi, and f2 are given in the following table. x fox) 1 0.1 2 0.7 3 0.1 4 fir) $2(:) 0.3 0.1 0.05 0.1 0.5 0.6 0.15 0.2 0.1 Let H3 = {HUH2}. In each of the following questions, use a randomized test if necessary. (a) (9 points) Does there exist a uniformly most powerful (UMP) size 0.05 test of Ho ver- sus Hz? If yes, then give the test and evaluate the power of your test under simple alternative hypothesis Hj. If no, then explain your reasoning. (b) (9 points) Does there exist a UMP size 0.2 test of Ho versus H3? If yes, then give the test and evaluate the power of your test under simple alternative hypothesis Hj. If no, then explain your reasoning. (c) (7 points) Does there exist a most powerful (MP) size 0.1 test of H3 versus Ho? If yes, then give the test and evaluate the power of your test under Ho. If no, then explain your reasoning