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

$$G _ { Y } ( t ) = \frac { t ^ { 2 } } { ( 4 - 3 t ) ^ { 4 } }$$

(a) Find $\mathrm { E } ( \mathrm { Y } )$.\\

(b) Write down an expression for the probability generating function of $X$.\\

(c) Find $\mathrm { P } ( X = 4 )$.\\

\hfill \mbox{\textit{CAIE Further Paper 4 2024 Q4}}