Stat 5101 Lecture Notes - School of Statistics

222 Stat 5101 (Geyer) Course NotesMomentsE(Y) =npvar(Y) =Mwhere M is the k × k matrix with elements{np i (1 − p i ),m ij =−np i p ji = ji ≠ jSpecializationdistributionThe special case n = 1 is called the multivariate BernoulliBer k (p) = Bin k (1, p)but for once we will not spell out the details with a special section for themultivariate Bernoulli. Just take n = 1 in this section.Marginal Distributions Distributions obtained by collapsing categories areagain multinomial (Section 5.4.5 in these notes).In particular, if Y ∼ Multi k (n, p), thenwhere(Y 1 ,...,Y j ,Y j+1 + ···+Y k )∼Multi j+1 (n, q) (B.4)q i = p i ,i ≤ jq j+1 = p j+1 + ···p kBecause the random vector in (B.4) is degenerate, this equation also givesimplicitly the marginal distribution of Y 1 , ..., Y jf(y 1 ,...,y j )()n=p y11y 1 ,...,y j ,n−y 1 −···−y ···pyj j (1 − p 1 −···−p j ) n−y1−···−yjjUnivariate Marginal DistributionsIf Y ∼ Multi(n, p), thenY i ∼ Bin(n, p i ).Conditional DistributionsIf Y ∼ Multi k (n, p), then(Y 1 ,...,Y j )|(Y j+1 ,...,Y k )∼Multi j (n − Y j+1 −···−Y k ,q),whereq i =p ip 1 + ···+p j,i =1,...,j.