Exercise 14.11 [LG-exercise]

Consider the family of linear Gaussian networks, as defined on page LG-network-page.

1. In a two-variable network, let $X_1$ be the parent of $X_2$, let $X_1$ have a Gaussian prior, and let ${\textbf{P}}(X_2X_1)$ be a linear Gaussian distribution. Show that the joint distribution $P(X_1,X_2)$ is a multivariate Gaussian, and calculate its covariance matrix.

2. Prove by induction that the joint distribution for a general linear Gaussian network on $X_1,\ldots,X_n$ is also a multivariate Gaussian.