Tower Property Verification in a Gaussian Markov Chain

Let $(X, Y, Z)$ be mean-zero jointly normal with $\operatorname{Var}(X) = \operatorname{Var}(Y) = \operatorname{Var}(Z) = 1$, $\operatorname{Corr}(X,Y) = 1/2$, $\operatorname{Corr}(Y,Z) = 1/3$, and $\operatorname{Corr}(X,Z) = 1/6$. (This makes $X - Y - Z$ a Gaussian Markov chain: $X \perp\!\!\perp Z \mid Y$.)

(a) Compute $E[X \mid Z]$ directly using the bivariate normal regression formula.

(b) Compute $E[E[X \mid Y] \mid Z]$ by first finding $E[X \mid Y]$, then taking its conditional expectation given $Z$.

(c) Verify that both answers agree, illustrating the tower property $E[X \mid Z] = E[E[X \mid Y] \mid Z]$ when $\sigma(Z) \subseteq \sigma(Y)$ is replaced by the Markov condition $X \perp\!\!\perp Z \mid Y$.