Consider the process of rolling a 3-sided dice n times, and let X1, X2, and X3 be discrete random variables that represe

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Consider The Process Of Rolling A 3 Sided Dice N Times And Let X1 X2 And X3 Be Discrete Random Variables That Represe 1 (99.63 KiB) Viewed 37 times

Consider the process of rolling a 3-sided dice n times, and let X1, X2, and X3 be discrete random variables that represent the number of times the respective number was rolled. The resulting probability mass function is, n! P(X1 = 21, X2 = 12, X3 = 13) 7p7pp3 21!x2!<3! with pi > 0 representing the unknown probability of face i being rolled, and the counts can take the values 21, 22, 23 € (0,1,2,...,n). Since each roll will result in exactly one face being rolled, it must also be that x1 + 12 +13 = n. Assume m random experiments are performed, each time rolling the dice n times. This results in the samples y = (21,1,21,2, 21,3), y2 = (22,1, 22,2, 22,3), ..., Ym = (xm,1, Xm,2, lm,3). Answer the following: (a) Noting that p3 = 1-pı-p2 and 21,3 = n-11,1–21,2 for all i = 1, 2,...,m, write down an expression for the log-likelihood function ((P1, P2). (b) We which to obtain the MLE for pı and p2 for this we need the multivari- ate score function (P1, P2) = ( pet ) and then solve for the maximum at S(P1, P2) = (0,0). Use this approach to show that, al al m m (P1, P2) = 1 11, 12 mn mn i=1 i=1 (c) Is (1, P2) an unbiased estimator? Note: E[Xj] = npi. (d) Is (P1, P2) a consistent estimator (challenging!). Note: Var[X] = np: (1 - Pi) and Cov[X, X;] = -np.p; for i j. (e) Obtain an expression for the for the Fisher Information Matrix, and hence form an expression for the approximate 95% confidence interval for Pı and p2. P1