Multivariate Gaussian Distribution

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.

2) whose distribution is given by (2) for p = 2. In this case it is customary to parametrize Σ (for reasons that will become clear) as follows: Σ = σ2 1 ρσ 1σ 2 ρσ 1σ 2 σ2 2 . Since detΣ = σ2 1 σ 2 2 (1−ρ 2) and detΣ > 0 (recall Σ is positive definite), we must have −1 < ρ < 1.

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  Distribution, Multivariate, Gaussian, Multivariate gaussian distributions

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