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Chapter 3 Random Vectors and Multivariate Normal …
sites.pitt.eduUncorrelated implies independence for multivariate normal random vari-ables 9. IfX, μ,andΣarepartitionedasabove, thenX1 andX2 areindependent if and only if Σ12 =0=ΣT 21. Proof. We will use m.g.f to prove this result. Two random vectors X1 and X2 are independent iff M(X 1,X2)(t1,t2)=MX 1 (t1)MX 2 (t2). Chapter 3 93
Lecture 1. Random vectors and multivariate normal …
www.stat.pitt.eduuniquely determined by the distributions of linear functions of t0X, for every t 2Rp. Corollary 4 paves the way to the de nition of (general) multivariate normal distribution. De nition 2. A random vector X2Rphas a multivariate normal distribution if t0Xis an univariate normal for all t 2Rp.