which is an affine combination of independent standard normal random vectors hencey is multivariate normal its expectation vector and covariance matrix can be found easily from theorem the next theorem shows that the distribution of subvector of multivariate normal random vector is again normal theorem marginal distributions of normal random vectors let (usbe an -dimensional normal random vector decompose xuand as up xp sr xus( uq xq sr sq where is the upper left corner of and sq is the lower right corner of thenx ( proofwe give proof assuming that is positive definite let bbbe the (lowercholesky decomposition of we can write #up xp bp uq xq cr cq { ( where and are independent pand -dimensional standard normal random vectors in particularx which means that ( )since > by relabeling the elements of we see that theorem implies that any subvector of has multivariate normal distribution for examplexq (uq sq the following theorem shows that not only the marginal distributions of normal random vector are normalbut also its conditional distributions theorem conditional distributions of normal random vectors let (usbe an -dimensional normal random vector with det( if is decomposed as in ( )then - ( xq (uq > - ( )sq sr sr as consequencex and xq are independent if and only if they are uncorrelatedthat isif sr (zero matrixprooffrom ( we see that + and xq uq +cr +cq consequently(xq uq cr - ( cq where is -dimensional multivariate standard normal random vector it follows that xq conditional on has (uq cr - ( )cq cq distribution the proof of