1 Consider the family of linear Gaussian networks, as defined in section 3.

a. In a two-variable network, let X1 be 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.