Chapter 3 Random Vectors and Multivariate Normal …

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Chapter 3Random Vectors and MultivariateNormal Random vectorsDefinition Random Vectors are Vectors of random83BIOS 2083Linear 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,..., 384BIOS 2083Linear 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 385BIOS 2083Linear ModelsAbdus S. WahedProperties of Mean and Covariance (cont.)

Bivariate normal distribution with mean (0,0)T and covariance matrix ... squares of independent normal random variables will be distributed indepen-dently as chi-squares. Chapter 3 94. BIOS 2083 Linear Models Abdus S. Wahed Theorem 3.3.2. If X∼ N

  Normal, Vector, Multivariate, Bivariate, Vectors and multivariate normal, Bivariate normal