theorem linear transformation if has an expectation vector and covariance matrix then the expectation vector of is uz ux ( and the covariance matrix of is ( ifin additiona is an invertible matrix and is continuous random vector with pdf then the pdf of the continuous random vector ax is given by (zf ( - zz rn det( )( where det( )denotes the absolute value of the determinant of proofwe have ez [axa ex au and [( )( ) [ ( )( ( )) [( )( )]aa afor invertible and continuous (as opposed to discrete)let ax and - consider the -dimensional cube [ hx [zn zn hthenp[ chn ( ) by definition of the joint density of let be the image of under - -that isall points such that ax recall from section that any matrix linearly transforms an -dimensional rectangle with volume into an -dimensional parallelepiped with volume det( )thusin addition to the above expression for [ ]we also have [ cp[ dhn det( - ) (xhn det( )|- (xequating these two expressions for [ ]dividing both sides by hn and letting go to we obtain ( for generalization of the linear transformation rule ( )consider an arbitrary mapping ( )written outg (xx (xgn (xxn