we write ( in and say that has standard normal distribution in rn let = +bz ( for some matrix and -dimensional vector then has expectation vector and covariance matrix bbsee ( and ( this leads to the following definition definition multivariate normal distribution an -dimensional random vector that can be written in the form ( for some -dimensional vector and matrix bwith ( in )is said to have multivariate normal or multivariate gaussian distribution with mean vector and covariance matrix bbwe write (usthe -dimensional density of multivariate normal distribution has very similar form to the density of the one-dimensional normal distribution and is given in the next theorem we leave the proof as an exercisesee exercise theorem density of multivariate random vector let (us)where the covariance matrix is invertible then has pdf ( - ( -us ( -um ( |sx rm ( figure shows the pdfs of two bivariate (that istwo-dimensionalnormal distributions in both cases the mean vector is [ ]and the variances (the diagonal elements of sare the correlation coefficients (orequivalently herethe covariancesare respectively and - - - - figure pdfs of bivariate normal distributions with means zerovariances and correlation coefficients (leftand (rightmultivariate normal