Problem 1 A data set (Ti, yi), for i = 1, ..., 50, is available in "HW11_data_1.csv" for the logistic regression model; P(Y. = j | 5, Di) = 1+ 3 ens' J = 1,2,3, and 1 P(Y. = 4 | 8, ) = 1+RIB, (1) (10 pts) Write down the log-likelihood function L(B) = log [ P(y: | B, I;). i=1 (2) (10 pts) Find the expressions of the partial derivatives OL 0- L aBj and aB;OBk for j = 1, 2, 3 and k = 1, 2, 3. (3) (10 pts) Using the partial derivatives just found, write and run a Newton-Raphson algorithm to obtain the maximum likelihood estimator 8. State the algorithm and the final result. (4) (10 pts) Find the predictive probabilities for y with a new predictor at r = r. (5) (10 pts) Hence, what would be the predicted outcome for y at this r