Transcription of Chapter 3 Random Vectors and Multivariate Normal …
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Chapter 3 Random Vectors and Multivariate Normal …
Chapter 3 Random Vectors and MultivariateNormal Random vectorsDefinition Random Vectors are Vectors of random83 BIOS 2083 Linear ModelsAbdus S. Wahedvariables. For instance,X= ,where each element represent a Random variable, is a Random Mean and covariance matrix of a Random mean (expectation) and covariance matrix of a Random vectorXis de-fined as follows:E[X]= E[X1]E[X2]..E[Xn] ,andcov(X)=E {X E(X)}{X E(X)}T = 21 1n 21 n1 2n ,( )where 2j=var(Xj)and jk=cov(Xj,Xk)forj, k=1,2,.., 384 BIOS 2083 Linear ModelsAbdus S. WahedProperties of Mean and IfXandYare Random Vectors andA,B,CandDare constant matrices,thenE[AXB+CY+D]=AE[X]B+CE[Y]+D. ( ) as an For any Random vectorX, the covariance matrixcov(X) is as an IfXj,j=1,2,..,nare independent Random variables, thencov(X)=diag( 2j,j=1,2,..,n). as an (X+a)=cov(X) for a constant as an 385 BIOS 2083 Linear ModelsAbdus S. WahedProperties of Mean and Covariance (cont.)
Left as an exercise. 2. For any random vector X, the covariance matrix cov(X) is symmetric. Proof. ... if and only if all non-zero linear combinations of the components of X are normally distributed. Chapter 3 90. ... Marginal and Conditional distributions
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