Question

1-1 postmultiply by . Take inverses of the resulting expression. -A22A21 I 4.13. Show the following if X is N,(. E) with 0. (a) Check that |E= |2 2||E,, - E, 2E . (Note that | can be factored into the product of contributions from the marginal and conditional distributions.) (b) Check that (x- μ ) Σ (x-μ) - [x1- μιΣι2 Σ7. (x2-μ.)]. - μ "Σ1,Σ3(x2 μ)1 x (Σ1-ΣΣΣ1 + (x -μ2), Σ( X2-μ) (Thus, the joint density exponent can be written as the sum of two terms corresponding to contributions from the conditional and marginal distributions.) (c) Given the results in Parts a and b, identify the marginal distribution of X2 and the conditional distribution of X,| X2 = x2.

Answer 1

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N-21229 221 Σ. Hemce, Provedl (b) let p-dimensienal vectort X - ( xuith Limilarly parto and let u be inn bane as yu u'and et E be paretitianiod as uwhene Ei is a submatuix with olilgx 9) 1 2 12 Z2 J Xhas multauati noumal duct with mear u and tve dlelinitu covauiounce matein 2 then a houmal auith meam u amd oaeiance amd E1 ndepedot & X aree spectiualy matrein E arol E2 astunption 2 0 Ondee the (a-μ) '(α -J (0) Z11 (2J .er 2 (- u) 2 2 (201-01) the (C) paxt Conpane fuomn

Fuom la), X) amd X 2E22X and X Nal)) amd X 22 indlepeudut aue Now, 2 = 1エ1 12lEl J) = lc-1| The joit dunaty e 12 -1 22 21 ー (1) +と12 -1 (2 2 (2) 22 1l2 12 Conditional pa )/2)

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