4 The discrete random variable \(X\) has probability generating function \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\) given by
$$G _ { X } ( t ) = 0.2 t + 0.5 t ^ { 2 } + 0.3 t ^ { 3 }$$
The random variable \(Y\) is the sum of two independent observations of \(X\).
- Find the probability generating function of \(Y\), giving your answer as an expanded polynomial in \(t\). [3]
- Use the probability generating function of \(Y\) to find \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\).