- - - ( is completed by observing that > - cr ( cr and - sq > - sr cq > sr cr cr cq cq cr |{zb cr if and xq are independentthen they are obviously uncorrelatedas sr [( )(xq uq ) ( (xq uq ) converselyif sr othen by ( the conditional distribution of xq given is the same as the unconditional distribution of xq that isn(uq sq in other wordsxq is independent of the next few results are about the relationships between the normalchi-squaredstudentand distributionsdefined in table recall that the chi-squared family of distributionsdenoted by kh are simply gamma( / / distributionswhere the parameter { is called the degrees of freedom kh distribution theorem relationship between normal and kh distributions if (usis an -dimensional normal random vector with det( then ( ) - ( ukh ( prooflet bbbe the cholesky decomposition of swhere is invertible since can be written as bzwhere [ zn ]is vector of independent standard normal random variableswe have - - ( us ( ( (bb ( uz zi = using the independence of zn the moment generating function of given by sy ( +***+zn [ sz szn sz pn = zi is where ( the moment generating function of is sz - / sz dz ( - ) dz ee  so that sy / which is the moment generating function of the gamma( / / distributionthat isthe kh distribution -see example the result now follows from the uniqueness of the moment generating function consequence of theorem is that if [ xn ]is -dimensional standard normalthen the squared length kxk xn has kh distribution if instead xi (ui ) then kxk is said to have noncentral kh distribution this distribution depends on the {ui only through the norm kuk we write kxk kh (th)where th kuk is the noncentrality parameter such distributions frequently occur when considering projections of multivariate normal random variablesas summarized in the following theorem noncentral kh distribution noncentrality parameter