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Answer the following questions: a. Consider the regression model 3;. 2 gm? + 6., for z' = 1,. . .,n, with ER.) 2 07 trade.) = 02, COU(,E;') = O for i j because 6., Ej are independent, and 61- N N (0, 0'). Show that the maximum likelihood estimate of 0 n 2 i=1 mi yr A 2 A - is 0 : zn . Find the distribution of 6. Suppose 02 is known. Using the distribution of 0 nd a 1 or 1:1 m? condence interval for 6. b. Suppose the observations (Y1, $1), . . . , (YH, 93,.) are collected, where w. is the size of an area and Y. is the number of certain plants in the area. Assume that the data are modeled as Poisson with Y. N Poisson(6:m) and the Yi's are independent. We want to estimate the parameter (9 using the method of least squares and the method of maximum likelihood. Find the estimates and their variances. c. Consider the model 3;. = [30 + 61:3.- + 61'. The Gauss-Markov conditions hold and also 6.4 N N (O, a). uppose we want to test the hypothesis Ho : 51 250 = 2 against the alternative that it is not true. Find the cov(l 2,80, 63) by expressing l 2% and 63 as a linear combinations of the K's. Explain the importance of this result. d. Refer to question (c). Find cov(31 2,30, 63) by using a method similar to cov(A+ B, 0+ D) = cov(A, C) +. . . Show all the details. e. Consider the model y. = )60 + 313:. + 61;. The Gauss-Markov conditions hold. Find the exact condition under which 8: