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 .Here, the argument of the exponential function, 12 2(x )2, is a quadratic function of thevariablex.
The concept of the covariance matrix is vital to understanding multivariate Gaussian distributions. Recall that for a pair of random variables X and Y, their covariance is defined as Cov[X,Y] = E[(X −E[X])(Y −E[Y])] = E[XY]−E[X]E[Y]. When working with multiple variables, the covariance matrix provides a succinct way to
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