4 The discrete random variable \(Y\) has probability generating function
$$\mathrm { G } _ { Y } ( t ) = 0.09 t ^ { 2 } + 0.24 t ^ { 3 } + 0.34 t ^ { 4 } + 0.24 t ^ { 5 } + 0.09 t ^ { 6 }$$
- Find the mean and variance of \(Y\).
\(Y\) is the sum of two independent observations of a random variable \(X\). - Find the probability generating function of \(X\), expressing your answer as a cubic polynomial in \(t\).
- Write down the value of \(\mathrm { P } ( X = 2 )\).