Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 46 / 1 / b / Solution

Take the coordinate-selection matrixThen $A^{T}X=X_1$, $A^T\mu ={\mu }_1$ and $A^T\Sigma A={\Sigma }_{11}$. The permitted result on a linear image of a multivariate normal vector therefore givesThis is the marginal distribution obtained by discarding the other coordinates. The conclusion holds also for a positive-semidefinite marginal covariance: in that case the Gaussian law may be degenerate and need not have a full-dimensional density. If the original covariance is positive definite, its principal block ${\Sigma }_{11}$ is positive definite as well.