Transcription of The Multivariate Gaussian Distribution
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The Multivariate Gaussian Distribution
The Multivariate Gaussian DistributionChuong B. DoOctober 10, 2008A vector-valued random variableX= X1 Xn Tis said to have amultivariatenormal (or Gaussian ) distributionwith mean Rnand covariance matrix Sn++1if its probability density function2is given byp(x; , ) =1(2 )n/2| |1/2exp 12(x )T 1(x ) .We write this asX N( , ). In these notes, we describe Multivariate Gaussians and someof their basic Relationship to univariate GaussiansRecall that the density function of aunivariate normal (or Gaussian ) distributionisgiven byp(x; , 2) =1 2 exp 12 2(x )2.
What do the isocontours of a multivariate Gaussian look like? As before, let’s consider the case where n = 2, and Σ is diagonal, i.e., x = x1 x2 µ = µ1 µ2 Σ = σ2 1 0 0 σ2 2 As we showed in the last section, p(x;µ,Σ) = 1 2πσ1σ2 exp − 1 2σ2 1 (x1 −µ1) 2 − 1 2σ2 2 (x2 −µ2) 2 . (4)
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