The random variable \(X\) is such that \(\mathrm{P}(X=r)=k r^{2}\) for \(r=1,2,3,4\), where \(k\) is a constant.
(a) Find the value of \(k\).

(b) Find the probability generating function \(\mathrm{G}_{X}(t)\) of \(X\).

The random variable \(Y\) has probability generating function \(\mathrm{G}_{Y}(t)=\frac{1}{4}+\frac{1}{2} t+\frac{1}{4} t^{2}\).
The random variable \(Z\) is the sum of \(X\) and \(Y\).
(c) Assuming that \(X\) and \(Y\) are independent, find the probability generating function \(\mathrm{G}_{Z}(t)\) of \(Z\) as a polynomial in \(t\).
(d) Given that \(\mathrm{E}(Z)=\frac{13}{3}\), use \(\mathrm{G}_{Z}(t)\) to find \(\operatorname{Var}(Z)\).