The discrete random variable \(X\) has probability generating function \(\mathrm{G}_{X}(t)\) given by
\(\mathrm{G}_{X}(t)=0.2 t+0.5 t^{2}+0.3 t^{3} .\)

The random variable \(Y\) is the sum of two independent observations of \(X\).
(a) Find the probability generating function of \(Y\), giving your answer as an expanded polynomial in \(t\).
(b) Use the probability generating function of \(Y\) to find \(\mathrm{E}(Y)\) and \(\operatorname{Var}(Y)\).