Each of the three random variables \(x, y\), and \(z\) has two levels: 0 and 1 . The joint distribution of these three variables can be determined from the facts \(\operatorname{Pr}(x=0, y=\) \(0, z=0)=\frac{1}{4}, \operatorname{Pr}(x=0, y=1, z=1)=\frac{1}{4}, \operatorname{Pr}(x=1, y=0, z=1)=\frac{1}{4}\), and \(\operatorname{Pr}(x=1, y=1, z=0)=\frac{1}{4}\).

(a) Are \(x, y\), and \(z\) mutually independent?

(b) Are \(x\) and \(y\) marginally independent?

(c) Are \(x\) and \(y\) independent given \(z\) ?

(d) Is \(x\) jointly independent of \(y\) and \(z\) ?