Stat 5101 Lecture Notes - School of Statistics

154 Stat 5101 (Geyer) Course Notesbecause X i X j = 0 with probability one.Hence var(X) =Mhas components{p i (1 − p i ),m ij =−p i p ji = ji ≠ j(5.37)We can also write this using more matrixy notation by introducing the diagonalmatrix P having diagonal elements p i and noting that the “outer product” pp ′has elements p i p j , hencevar(X) =P−pp ′Question: Is var(X) positive definite? This is of course related to the questionof whether X is degenerate. We haven’t said anything explicit about either,but the information needed to answer these questions is in the text above. Itshould be obvious if you know what to look for (a good exercise testing yourunderstanding of degenerate random vectors).5.4 The Multinomial DistributionThe multinomial distribution is the multivariate analog of the binomial distribution.It is sort of, but not quite, the multivariate generalization, that is, thebinomial distribution is sort of, but not precisely, a special case of the multinomialdistribution. Thus is unlike the normal, where the univariate normaldistribution is precisely the one-dimensional case of the multivariate normal.Suppose X 1 , X 2 are an i. i. d. sequence of Ber k (p) random vectors (caution:the subscripts on the X i indicate elements of an infinite sequence of i. i. d.random vectors, not components of one vector). ThenY = X 1 + ···+X nhas the multinomial distribution with sample size n and dimension k, abbreviatedY ∼ Multi k (n, p)if we want to indicate the dimension in the notation or just Y ∼ Multi(n, p) ifthe dimension is clear from the context.Note the dimension is k, not n, that is, both Y and p are vectors of dimensionk.5.4.1 Categorical Random VariablesRecall that a multinomial random vector is the sum of i. i. d. BernoullisY = X 1 + · + X nand that each Bernoulli is related to a categorical random variable: X i,j =1ifand only if the i-th observation fell in the j-th category. Thus Y j = ∑ i X i,j isthe number of individuals that fell in the j-th category.