The random variable \(X\) has probability generating function \(\mathrm{G}_{X}(t)\) given by
\(\mathrm{G}_{X}(t)=k\left(1+3 t+4 t^{2}\right)\)
where \(k\) is a constant.
(a) Show that \(\mathrm{E}(X)=\frac{11}{8}\).

The random variable \(Y\) has probability generating function \(\mathrm{G}_{Y}(t)\) given by
\(\mathrm{G}_{Y}(t)=\frac{1}{3} t^{2}(1+2 t)\)

The random variables \(X\) and \(Y\) are independent and \(Z=X+Y\).
(b) Find the probability generating function of \(Z\), expressing your answer as a polynomial in \(t\).

(c) Use your answer to part (b) to find the value of \(\operatorname{Var}(Z)\).

(d) Write down the most probable value of \(Z\).