Two dice are thrown. Their respective random outcomes are (X_{1}) and (X_{2}). Let (X=max left(X_{1}, X_{2} ight))

Two dice are thrown. Their respective random outcomes are \(X_{1}\) and \(X_{2}\). Let \(X=\max \left(X_{1}, X_{2}\right)\) and \(Y\) be the number of even components of \(\left(X_{1}, X_{2}\right) . X\) and \(Y\) have the respective ranges \(R_{X}=\{1,2,3,4,5,6\}\) and \(R_{Y}=\{0,1,2\}\).

(1) Determine the joint probability distribution of the random vector \((X, Y)\) and the corresponding marginal distributions. Are \(X\) and \(Y\) independent?

(2) Determine \(E(X), E(Y)\), and \(E(X Y)\).