Three-Level Normal Hierarchy: Iterated Tower and Smoothing

Consider a three-level normal hierarchy: $Z \sim N(0, 1)$, then $Y \mid Z \sim N(Z, 1)$, then $X \mid Y \sim N(Y, 1)$.

(a) Using iterated expectations, find $E[X]$ and $\operatorname{Var}(X)$.

(b) Find $E[X \mid Z]$ by applying the tower property: $E[X \mid Z] = E[E[X \mid Y] \mid Z]$.

(c) Verify part (b) by computing $\operatorname{Cov}(X, Z)$ and using the joint normality of $(X, Z)$.