The following table shows the probability distribution of the discrete random variable $X$.

(a)     Show that the probability generating function of $X$ is given by

$$G(t) = \frac{{t{{(1 + t)}^2}}}{4}.$$

(b)     Given that $Y = {X_1} + {X_2} + {X_3} + {X_4}$, where ${X_1},{\text{ }}{X_2},{\text{ }}{X_3},{\text{ }}{X_4}$ is a random sample from the distribution of $X$,

(i)     state the probability generating function of $Y$;

(ii)     hence find the value of ${\text{P}}(Y = 8)$.

(a)     $G(t) = \frac{1}{4}t + \frac{1}{2}{t^2} + \frac{1}{4}{t^3}$     M1A1

$= \frac{{t{{(1 + t)}^2}}}{4}$     AG

[2 marks]

(b)     (i)     ${\text{PGF of }} Y = {\left( {G(t)} \right)^4}\left( { = {{\left( {\frac{{t{{(1 + t)}^2}}}{4}} \right)}^4}} \right)$     A1

(ii)     ${\text{P}}(Y = 8) = {\text{coefficient of }}{t^8}$     (M1)

$= \frac{{^8{{\text{C}}_4}}}{{256}}$     (A1)

$= \frac{{35}}{{128}}   (0.273)$     A1

Note: Accept $0.27$ or answers that round to $0.273$.

[4 marks]