| Date | May 2018 | Marks available | 4 | Reference code | 18M.3sp.hl.TZ0.2 |
| Level | HL only | Paper | Paper 3 Statistics and probability | Time zone | TZ0 |
| Command term | Show that | Question number | 2 | Adapted from | N/A |
Question
Consider an unbiased tetrahedral (four-sided) die with faces labelled 1, 2, 3 and 4 respectively.
The random variable X represents the number of throws required to obtain a 1.
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.
Markscheme
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]
