2 The random variable \(X\) has probability generating function \(\mathrm { G } _ { X } ( t )\) given by $$\mathrm { G } _ { X } ( t ) = \frac { 1 } { 5 } + p t + q t ^ { 2 }$$ where \(p\) and \(q\) are constants.

1. Given that \(\mathrm { E } ( X ) = 1.1\), find the numerical value of \(\operatorname { Var } ( X )\).
   \includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-04_2714_38_109_2010} The random variable \(Y\) has probability generating function \(\mathrm { G } _ { Y } ( t )\) given by $$\mathrm { G } _ { Y } ( t ) = \frac { 2 } { 3 } t \left( 1 + \frac { 1 } { 2 } t ^ { 2 } \right)$$ The random variable \(Z\) is the sum of independent observations of \(X\) and \(Y\).
2. Find the probability generating function of \(Z\).
3. Find \(\mathrm { P } ( Z > 2 )\).
4. State the most probable value of \(Z\).