Show that the probability generating function for $X$ is given by $G(t) = \frac{P}{{1 - q{t^2}}}$.

Hence determine ${\text{E}}(X)$ in terms of $p$ and $q$.

The random variable $Y$ is given by $Y = 2X + 1$. Find the probability generating function for $Y$.

$G(t) = \sum {P(X = x){t^x}}$     (M1)

$= p + pq{t^2} + p{q^2}{t^4} +  \ldots$

(summing $GP$) ${u_1} = p,{\text{ }}r = q{t^2}$     A1

$= \frac{p}{{1 - q{t^2}}}$     AG

[2 marks]

$G’(t) =  - \frac{p}{{{{(1 - q{t^2})}^2}}} \times  - 2qt$     M1A1

${\text{E}}(X) = G’(1)$     (M1)

$= \frac{{2pq}}{{{{(1 - q)}^2}}}\,\,\,\left( { = \frac{{2q}}{p}} \right)$     A1

[4 marks]

METHOD 1

${\text{PGF of }}Y = \sum {P(Y = y){t^y}}$     (M1)

$= pt + pq{t^5} + p{q^2}{t^9} +  \ldots$     A1

$= \frac{{pt}}{{1 - q{t^4}}}$     A1

METHOD 2

${\text{PGF of }}Y = {\text{E}}({t^Y})$     (M1)

$= {\text{E}}({t^{2X + 1}})$

$= {\text{E}}\left( {{{({t^2})}^X}} \right) \times {\text{E}}(t)$     A1

$= \frac{{pt}}{{1 - q{t^4}}}$     A1

[3 marks]