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2. Consider two independent random samples X1, .... Xn and Yl, ..., Ym, and both of them follow normal distributions. More specifically, Xi ~ N(#1, o') and Y; ~ N(12, 0?), where #1, #2 and o' are unknown parameters. Suppose we wish to test the hypothesis Ho : /1 = /2 v.s. H1 : /1 / /2. Let 1(#1, /2) denote the log-likelihood function from these two samples. (a) Show that the log-likelihood function is given by 1(/1, 12) = -=(n + m) logo? - 202 (Xi - M1 )2 + _(Y; - 12)2). j=1 (b) Hence, derive the maximum likelihood estimators of /1, /2 and o2. (c) Derive the likelihood ratio test statistic for testing the hypothesis specified above. (d) The, specify the rejection region given by the likelihood ratio test