| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2024 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Generating Functions |
| Type | Solve for parameters using PGF coefficients |
| Difficulty | Challenging +1.2 This is a standard Further Maths probability generating functions question requiring routine application of PGF properties: using G'(1) for expectation, G''(1) for variance, multiplying PGFs for independent sums, and reading coefficients. While it involves multiple parts and careful algebra, all techniques are textbook exercises with no novel insight required. The topic itself (PGFs) is Further Maths content, placing it above average difficulty on an absolute scale, but within Further Maths it's straightforward. |
| Spec | 5.02a Discrete probability distributions: general |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \([G_X(t) = \frac{1}{5} + pt + qt^2,\quad G'_X(t) = p + 2qt,]\quad p + 2q = 1.1\) | B1 | |
| \(\frac{1}{5} + p + q = 1,\quad p + q = \frac{4}{5}\) and solve | M1 | |
| \(\left[p = \frac{1}{2},\right]\quad q = \frac{3}{10}\) | A1 | |
| \(G''_X(t) = 2q = \frac{3}{5}\), \(\text{Var}(X) = 0.6 + 1.1 - 1.1^2 = 0.49\) | A1 | |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| \(G_Z(t) = \left(\frac{1}{5}+\frac{1}{2}t+\frac{3}{10}t^2\right)\times\frac{2}{3}t\left(1+\frac{1}{2}t^2\right)\) | M1 | Wrong \(p\) and \(q\) (or swapped) or missing \(t\), M1A0. \(p\) and \(q\) must have their numerical values from part (a). |
| \(\frac{1}{30}\left(4t+10t^2+8t^3+5t^4+3t^5\right)\) | A1 | Accept answer in any equivalent form. |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{16}{30}=\frac{8}{15}\) oe | B1 FT | FT their fully expanded form of PGF. |
| 1 |
| Answer | Marks | Guidance |
|---|---|---|
| \((Z=)\ 2\) | B1 | Correct work only, must see correct fully expanded polynomial form of PGF. |
| 1 |
## Question 2(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[G_X(t) = \frac{1}{5} + pt + qt^2,\quad G'_X(t) = p + 2qt,]\quad p + 2q = 1.1$ | B1 | |
| $\frac{1}{5} + p + q = 1,\quad p + q = \frac{4}{5}$ and solve | M1 | |
| $\left[p = \frac{1}{2},\right]\quad q = \frac{3}{10}$ | A1 | |
| $G''_X(t) = 2q = \frac{3}{5}$, $\text{Var}(X) = 0.6 + 1.1 - 1.1^2 = 0.49$ | A1 | |
| **Total: 4** | | |
## Question 2(b):
$G_Z(t) = \left(\frac{1}{5}+\frac{1}{2}t+\frac{3}{10}t^2\right)\times\frac{2}{3}t\left(1+\frac{1}{2}t^2\right)$ | M1 | Wrong $p$ and $q$ (or swapped) or missing $t$, M1A0. $p$ and $q$ must have their numerical values from part (a).
$\frac{1}{30}\left(4t+10t^2+8t^3+5t^4+3t^5\right)$ | A1 | Accept answer in any equivalent form.
| 2 |
## Question 2(c):
$\frac{16}{30}=\frac{8}{15}$ oe | B1 FT | FT their fully expanded form of PGF.
| 1 |
## Question 2(d):
$(Z=)\ 2$ | B1 | Correct work only, must see correct fully expanded polynomial form of PGF.
| 1 |
---2 The random variable $X$ has probability generating function $\mathrm { G } _ { X } ( t )$ given by
$$\mathrm { G } _ { X } ( t ) = \frac { 1 } { 5 } + p t + q t ^ { 2 }$$
where $p$ and $q$ are constants.
\begin{enumerate}[label=(\alph*)]
\item Given that $\mathrm { E } ( X ) = 1.1$, find the numerical value of $\operatorname { Var } ( X )$.\\
\includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-04_2714_38_109_2010}
The random variable $Y$ has probability generating function $\mathrm { G } _ { Y } ( t )$ given by
$$\mathrm { G } _ { Y } ( t ) = \frac { 2 } { 3 } t \left( 1 + \frac { 1 } { 2 } t ^ { 2 } \right)$$
The random variable $Z$ is the sum of independent observations of $X$ and $Y$.
\item Find the probability generating function of $Z$.
\item Find $\mathrm { P } ( Z > 2 )$.
\item State the most probable value of $Z$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2024 Q2 [8]}}