| Date | May 2014 | Marks available | 6 | Reference code | 14M.1.hl.TZ0.3 |
| Level | HL only | Paper | 1 | Time zone | TZ0 |
| Command term | Find, Hence, Show that, and State | Question number | 3 | Adapted from | N/A |
Question
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 \({\text{P}}(Y = 8)\).
Markscheme
(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]
