2. Let T = (71, . . . , Wk) be a probability vector, i.e. 7, 2 0 for all j = 1, . . ., k, Zin, = 1. Let pr denote the statistical model Y1, . .. , Yn ~ with sample size n. (a) Write out the log-likelihood lT (Y1, . . . , Yn) = logp. (Y1, . . . , Yn). (b) Let (71, . . . , Wk-1) be the free parameters, and 7 = 1 - _._7j. Show that the score function S, = ($7,1, . . . , ST,k-1) is given by n ST,j (Y1, . . . , Yn) = E 1(Yi = j) 1( Yi = k) TK i=1 (c) Verify that the Fisher information matrix I(7) is given by 11T I(7) = n diag(] , . . . + TT K where 1 E RK- is the column vector consisting of all ones. (d) Using the Woodbury matrix identity (consult wikipedia), compute I(7)-1. Com- pare your result with your answer to 2(a) in HW2. What do you find