Stat 5101 Lecture Notes - School of Statistics

160 Stat 5101 (Geyer) Course NotesProof of Theorem 5.17. Just calculate. The relevant marginal is the distributionof (Y j+1 ,...,Y k ) but that isn’t a brand name distribution. Almost as goodis the marginal ofZ =(Y 1 +···+Y j ,Y j+1 ,...,Y k )=(n−Y j+1 −···−Y k ,Y j+1 ,...,Y k ) (5.40)which is Multi k−j+1 (n, q) withq =(p 1 +···+p j ,p j+1 ,...,p k )=(n−p j+1 −···−p k ,p j+1 ,...,p k )It’s almost the same thing really, because the right hand side of (5.40) is afunction of Y j+1 , ..., Y k alone, henceP (Y i = y i ,i=j+1,...,k)()n=n−y j+1 −···−y k ,y j+1 ,...,y k×(1 − p j+1 −···−p k ) n−yj+1−···−y kp yj+1j+1 ···py kkAnd, of course, conditional equals joint over marginal( )ny 1,...,y kpy 11 ···py kk( )nn−y j+1−···−y k ,y j+1,...,y k(1 − pj+1 −···−p k ) n−yj+1−···−y yk pj+1j+1 ···py kkn!=y 1 !···y k ! · (n−y j+1 −···−y k )!y j+1 ! ···y k !n!p y11×···pyj j(1 − p j+1 −···−p k ) n−yj+1−···y k= (n − y j+1 −···−y k )!y 1 ! ···y j !j∏() yjp i1 − pi=1 j+1 −···p k( ) j∏ (n−yj+1 −···−y k=y 1 ,...,y ji=1and that’s the conditional density asserted by the theorem.p ip 1 + ···+p j) yjProblems5-1. Is ⎛⎞⎝ 3 2 −12 3 2 ⎠−1 2 3a covariance matrix? If not, why not?