Find component PGF from sum PGF

2 The probability generating function, \(\mathrm { G } _ { Y } ( t )\), of the random variable \(Y\) is given by $$G _ { Y } ( t ) = 0.04 + 0.2 t + 0.37 t ^ { 2 } + 0.3 t ^ { 3 } + 0.09 t ^ { 4 }$$

1. Find \(\operatorname { Var } ( Y )\).
   The random variable \(Y\) is the sum of two independent observations of the random variable \(X\).
2. Find the probability generating function of \(X\), giving your answer as a polynomial in \(t\).

4 The random variable \(Y\) is the sum of two independent observations of the random variable \(X\). The probability generating function \(\mathrm { G } _ { Y } ( \mathrm { t } )\) of \(Y\) is given by $$G _ { Y } ( t ) = \frac { t ^ { 2 } } { ( 4 - 3 t ) ^ { 4 } }$$

1. Find \(\mathrm { E } ( \mathrm { Y } )\).
2. Write down an expression for the probability generating function of \(X\).
3. Find \(\mathrm { P } ( X = 4 )\).

4 The discrete random variable \(Y\) has probability generating function $$\mathrm { G } _ { Y } ( t ) = 0.09 t ^ { 2 } + 0.24 t ^ { 3 } + 0.34 t ^ { 4 } + 0.24 t ^ { 5 } + 0.09 t ^ { 6 }$$

1. Find the mean and variance of \(Y\).
   \(Y\) is the sum of two independent observations of a random variable \(X\).
2. Find the probability generating function of \(X\), expressing your answer as a cubic polynomial in \(t\).
3. Write down the value of \(\mathrm { P } ( X = 2 )\).