2 The probability generating function, $\mathrm { G } _ { Y } ( t )$, of the random variable $Y$ is given by

$$G _ { Y } ( t ) = 0.04 + 0.2 t + 0.37 t ^ { 2 } + 0.3 t ^ { 3 } + 0.09 t ^ { 4 }$$

(a) Find $\operatorname { Var } ( Y )$.\\

The random variable $Y$ is the sum of two independent observations of the random variable $X$.\\
(b) Find the probability generating function of $X$, giving your answer as a polynomial in $t$.\\

\hfill \mbox{\textit{CAIE Further Paper 4 2022 Q2}}