2. (Chapter 2, pp. 100-101) Consider the simple linear regression model: 3;;- = 50 + 1313:;- + 61:. If we assume a; ~ N (0, 02), we have y ~ N (/30 + {31%, 0'2). The likelihood function is constructed as 1 exp V 211172 L(5o,l31,02) = H ( e\". a, am\"). (20 points) i=1 (a) Show that we have n(L) a m(L(o,1,cr2)) = '2 ln27r 211102 % Em so 5m)? i=1 (5 points) (b) For a xed 02 > 0, maximizing ln(L) is equivalent to minimizing 2W: [in lm-)2. Show that the 1:1 maximum likelihood estimators of g and l are 50 = 37 311?, and 51 = %, respectively. (10 points) 1:1: (c) Next, maximizing ln(L(0, 31,02 with respect to 0'2 is equivalent to minimizing n n 2 1 \" " 2 - 2 E inc + 27:2 1:1:(19; 50 [31331) , With respect to 0' . Z?=1(yi 30 519302 TL Show that the maximum likelihood estimator of 0'2 is . (5 points)