(b) Show that if face) = i/:Wexp _%:c2 ,xER Mm) = exp(_|wp, mere, 2 then the rejection region for the Uniformly Most Powerful (UMP) level 0: test of H0 against H1 is given by 11 ('wl _ l)2 > k for some I: that depends on 0:. (Note that it is not necessary to nd a closed form of k in terms of a. Describing mathematical relationship between k and a should be sufficient). (c) Calculate both probability of type I error and probability of type II error for the test in part (b) when n = 1 and (i) k = 1 or (ii) k _ :1. Present the results to three digits beyond the decimal point. (You may use calculator, R, or table of standard normal distribution if needed). ((1) Determine whether the two tests in (c) are unbiased or not. 3. Let X1, . . . ,X be an i-i-d- random sample from the following pdf 1 f(:c'6)= E3326";a 0 60 . (b) Using the fact2 \"1.9 a N x03\") construct the level 0: UMP test of H: 66 agamstfz: =61whenn=4,6=1,16=3,anda=0.05. Calculate the power of this UMP test. Present the results to three digits beyond the decimal point. (You may use calculator, R, or table of chi squared distribution if needed). (c) Find the level a UMP test for H0 :6 6D (for arbitrary 71,60,05) ((1) Find a size a LRT for H0 : 6 60 based on sufficient statistics. Justify whether the LET is the level 0: UMP test or not