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.)
and the random variables are said to be exchangeable. 3.2 Multivariate Normal Distribution Definition 3.2.1. Multivariate Normal Distribution. A random vector X =(X1,X2,...,X n) T is said to follow a multivariate normal distribution with mean μ and covariance matrix Σ if X canbeexpressedas X= AZ+μ, where Σ= AAT and Z=(Z1,Z2,...,Z n) with Z
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