proofthe cdf of is given by [ zp[( )/ zp[ szz +sz - dx - / dy ph( ) where we make change of variable ( )/ in the fourth equation hencez ( the rescaling procedure in theorem is called standardization it follows from theorem that any (us can be written as szstandardization where ( in other wordsany normal random variable can be viewed as an affine transformation -that isa linear transformation plus constant -of standard normal random variable we now generalize this to dimensions let zn be independent and standard normal random variables the joint pdf of [ zn ]is given by fz (ze zi ( ) = rn affine transformation ( we write ( )where is the identity matrix consider the affine transformation = +bz ( for some matrix and -dimensional vector note thatby ( and ( ) has expectation vector and covariance matrix bbwe say that has multivariate normal or multivariate gaussian distribution with mean vector and covariance matrix we write (usthe following theorem states that any affine combination of independent multivariate normal random variables is again multivariate normal theorem affine transformation of normal random vectors let xr be independent mi -dimensional normal random vectorswith xi (ui si ) thenfor any vector and mi matrices br abi xi bi ui bi si > ( = = = proofdenote the -dimensional random vector in the left-hand side of ( by by definitioneach xi can be written as ui ai where the { are independent (because the {xi are independent)so that ar = bi (ui ai = bi ui = bi ai multivariate normal