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Question 3: Let X be a random variable with a normal distribution N(M1, oî), and Y a random variable with a normal distribution N(M2, oầ). Consider a random sample of size nį from X and a random sample of size n2 from Y. Assume that X and Y are independent. = = (a) Assuming that oỉ = oź = oº, show that the maximum likelihood estimator for o? is 22 - (n1 – 1)sî + (n2 – 1)sî ni + n2 + where 2121(X; - X) ni - 1 and sa 272 (Y; -Y) n2 - 1 (b) Assuming that oỉ + ož, show that the maximum likelihood estimators for oị and oz are: o (n1 - 1)s (n2 – 1)sź n1 n2 (c) Let Lo be the likelihood function evaluated at the maximum likelihood estimators of o2, M1 and Ma in part (a) and Ly be the likelihood function evaluated at the maximum likelihood estimators of oſ, oż, and M2 in part (b). To test the hypothesis H. : 01 = 02 against H1 :01 # 02, we use the likelihood ratio A = L/L1. Show that (67)1/2 (62)^2/2 A= (@2)(nı+n2)/2