The discrete random variable \(X\) has probability generating function \(G_X(t)\) given by

\(G_X(t)=\frac{t}{(3-2t)^2}\).

(a) Find \(\mathrm{E}(X)\) and \(\operatorname{Var}(X)\).

The discrete random variable \(Y\) has probability generating function \(G_Y(t)\) given by

\(G_Y(t)=\frac{t^2}{(3-2t)^2}\).

The random variable \(Z\) is the sum of the random variables \(X\) and \(Y\).

(b) Assuming \(X\) and \(Y\) are independent, find \(\mathrm{P}(Z\gt 4)\).