Consider three random variables, \(X, Y\), and \(Z\). Suppose that \(Y\) takes on \(k\) values \(y_{1}, \ldots, y_{k}\); that \(X\) takes on \(l\) values \(x_{1}, \ldots, x_{l}\); and that \(Z\) takes on \(m\) values \(z_{1}, \ldots, z_{m}\). The joint probability distribution of \(X, Y, Z\) is \(\operatorname{Pr}(X=x, Y=y, Z=z)\), and the conditional probability distribution of \(Y\) given \(X\) and \(Z\) is \(\operatorname{Pr}(Y=y \mid X=x, Z=z)=\frac{\operatorname{Pr}(Y=y, X=x, Z=z)}{\operatorname{Pr}(X=x, Z=z)}\).

a. Explain how the marginal probability that \(Y=y\) can be calculated from the joint probability distribution. This is a generalization of Equation (2.16).

b. Show that \(E(Y)=E[E(Y \mid X, Z)]\). This is a generalization of Equations (2.19) and (2.20).