38. In the multinomial model (38), the maximum-likelihood estimators Pi of the P's are Pi = x Jn. [The following are two methods for proving this result: (i) Maximize log P( XI' . . . , x m ) subject to 'E. Pi = 1 by the method of undetermined multipliers. (ii) Show that npt· n(xJny' by considering n numbers of which X i are equal to pJx; for i = 1,.. . , m and noting that their geometric mean is less than or equal to their arithmetic mean.)