2. Consider two independent random samples Xl, . n and Y 1, , m, and both of them follow normal distributions. More specifically, Xi N (111, 02) and Y) N (112, 02), where VI, and 02 are unknown parameters. Suppose we wish to test the hypothesis 110 : III V.S. # 112. Let 1011, 112) denote the log-likelihood function from these two samples. (a) (b) (c) (d) Show that the log-likelihood function is given by 1011, '12) 1 1 (n + m) log 02 20-2 2 1 /12)2 Hence, derive the maximum likelihood estimators of VI, and 02. Derive the likelihood ratio test statistic for testing the hypothesis specified above. The, specify the rejection region given by the likelihood ratio test.