Consider the family of linear Gaussian networks, as illustrated.

a. In a two-variable network, let X1 he the parent of X2, let X1 have a Gaussian prior, and let P (X2, X1) be a linear Gaussian distribution. Show that the joint distribution P(X1, X2) is a multivariate Gaussian, and calculate its covariance matrix.

b. Prove by induction that the joint distribution for a general linear Gaussian network on X1..., Xn, is also a multivariate Gaussian.