{"word_count": 102, "character_count": 676}
This morning, in our mathematical statistical class, we've seen briefly the multinomial distribution, and statistical inference.  has a  distribution if its probability function iswith  and .The maximum likelihood estimator is then the optimum ofWe use Lagrange multiplier to solve this constrained optimization problem,First order conditions are hereandThus,Fromwe can easily get that Lagrande multiplier is . And thenOne can easily get that this maximum likelihood estimator is unbiased, since . Actually, we can easily prove thatand that , while . The trick to get the later is simple,and . Thus, we can easily get the covariance. From that term, we can write thatwithwhile