THRY PROBAB &STAT I
THRY PROBAB &STAT I STAT 542
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This 1 page Class Notes was uploaded by Giovani Ullrich PhD on Saturday September 26, 2015. The Class Notes belongs to STAT 542 at Iowa State University taught by Staff in Fall. Since its upload, it has received 58 views. For similar materials see /class/214403/stat-542-iowa-state-university in Statistics at Iowa State University.
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Date Created: 09/26/15
Summary of Some Useful Facts About Multivariate Normal Distributions 1 If X has mean vector u and covariance matrix 2 then Y B X d has mean vector kxl kxk 1X1 lxk kxl 1X1 B u d and covariance matrix B 2 Bi 1X1 lxk M lxk kxk kxl 2 The MVN distribution is most usefully de ned as the distribution of g kA Z u for Z a gtlt1 Xp pxl kxl le vector of independent standard normal random variables Such a random vector has mean vector u and covariance matrix 13 kA A This de nition turns out to be unambiguous M x xp px any dimension p and any matrixA giving a particularZ end up producing the same k dimensional joint distribution 3 IinSMVNkJ thhen Y BXd is MVN1B u 513 2 B kgtlt1 kxl kxk 1X1 lxk kgtlt1 1X1 lxk kxl 1X1 lxk kxk kxl 4 If X is MVNk its individual marginal distributions are univariate normal Further any sub vector of dimensionl lt k is MVNZ with mean vector the appropriate subvector of u and kxl covariance matrix the appropriate submatrix of Z kxk 5 If g is MVNk Mlnjand independent of Y which is MVNZ 1222 then the vector X1 kxl kxk X1 1x1 1x1 X it 21 W 1s MVN 1H 1 Mm Y L5 0 222 1x1 W 1x1 6 For nonsingular ka the MVNk Age distribution has a joint pdf on kdimensional space X kxl X given by i 1 1 i 1 fXx 27239 2 ldetZl 2 exp Ex u Z x y 7 The joint pdf given in 6 above can be studied and conditional distributions given values for 1 part ofthe Xvector identi ed For kX M MVNk Majwhere X1 kx gtlt krl2gtlt1 1 11 212 XI an XI MM 1 ILLZ ka 221 222 MN krlgtltl buxom the conditional distribution of X 1 given that X 2 x2 is MVNZ with mean vector u1 2122 xi LLZ and covariance matrix 211 21222221 8 All correlations between two parts of a MVN vector equal to 0 implies that those parts of the vector are independent