Transcription of Lecture 1. Random vectors and multivariate normal …
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Lecture 1. Random vectors and multivariate normal …
Lecture 1. Random vectors and multivariate normal Moments of Random vectorA Random vectorXof sizepis a column vector consisting ofprandom variablesX1,..,Xpand isX= (X1,..,Xp) . The mean or expectation ofXis defined by the vector ofexpectations, E(X) = E(X1)..E(Xp) ,which exists ifE|Xi|< for alli= 1,.., a Random vector of sizepandYbe a Random vector of sizeq. Forany non- Random matricesA(m p),B(m q),C(1 n), andD(m n),E(AX+BY) =AE(X) +BE(Y),E(AXC+D) =AE(X)C+ a Random vectorXof sizepsatisfyingE(X2i)< for alli= 1,..,p, the variance covariance matrix (or just covariance matrix) ofXis Cov(X) =E[(X EX)(X EX) ].The covariance matrix ofXis ap psquare, symmetric matrix. In particular, ij=Cov(Xi,Xj) = Cov(Xj,Xi) = properties:1. Cov(X) =E(XX ) E(X)E(X) .2. Ifc=c(p 1)is a constant, Cov(X+c) = Cov(X).3. IfA(m p)is a constant, Cov(AX) =ACov(X)A .Lemma pmatrix is a covariance matrix if and only if it is multivariate normal distribution - nonsingular caseRecall that the univariate normal distribution with mean and variance 2has densityf(x) = (2 2) 12exp[ 12(x ) 2(x )].
De nition 1. Let 2Rp and (p p) >0. A random vector X2R p has p-variate normal distribution with mean and covariance matrix if it has probability density function f(x) = j2ˇ 0j12 exp 1
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