Multivariate Gaussian Distribution - Mathematics Home

Multivariate Gaussian DistributionThe random vectorX= (X1,X2,..,Xp)is said to have amultivariate Gaussian distributionif the joint distributionofX1,X2,..,Xphas densityfX(x1,x2,..,xp) =1(2 )p/2det( )1/2exp( 12(x )t 1(x ))(1)where is ap psymmetric, positive definite matrix. The notation is asfollows:xis the column vectorx= , is the column vector = 1 p , 1is the inverse of the matrix andtdenotes matrix transposition. Thusthe quantity appearing in the exponential is a 1 pmatrix times ap pmatrix times ap 1 matrix; and hence, a 1 1 matrix, a real (x )t 1(x ) =p k,`=1(xk k) 1k`(x` `)where 1k`is the (k,`)th matrix element of 1. The constants in front ofthe exponential are normalization constants; that is, if (1) is integrated overRpthen the result equals 1.

Multivariate Gaussian Distribution The random vector X = (X 1,X 2,...,X p) is said to have a multivariate Gaussian distribution if the joint distribution of X 1,X 2,...,X p has density f X(x 1,x 2,...,x p) = 1 (2π)p/2 det(Σ)1/2 exp − 1 2 (x−µ)tΣ−1(x−µ) (1) follows: x is the column vector x = x 1 x 2... x p , µ is the column vector ...

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  Distribution, Multivariate, Gaussian, Multivariate gaussian distributions

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