State the distribution of X.

Show that the probability generating function, $G\left( t \right)$, for X is given by $G\left( t \right) = \frac{t}{{4 - 3t}}$.

Find $G'\left( t \right)$.

Determine the mean number of throws required to obtain a 1.

X is geometric (or negative binomial)      A1

[1 mark]

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

recognition of GP $\left( {{u_1} = \frac{1}{4}t,\,\,r = \frac{3}{4}t} \right)$     (M1)

$= \frac{{\frac{1}{4}t}}{{1 - \frac{3}{4}t}}$     A1

leading to $G\left( t \right) = \frac{t}{{4 - 3t}}$     AG

[4 marks]

attempt to use product or quotient rule      M1

$G'\left( t \right) = \frac{4}{{{{\left( {4 - 3t} \right)}^2}}}$     A1

[2 marks]

4      A1

Note: Award A1FT to a candidate that correctly calculates the value of $G'\left( 1 \right)$ from their $G'\left( t \right)$.

[1 mark]