| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2023 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Generating Functions |
| Type | Determine constant in PGF |
| Difficulty | Standard +0.8 This is a multi-part Further Maths statistics question on PGFs requiring: (a) finding k using G(1)=1 then computing E(X)=G'(1), (b) multiplying two PGFs and expanding, (c) using G''(1) to find Var(Z), (d) identifying the mode from coefficients. While systematic, it requires fluency with PGF properties, algebraic manipulation, and multiple connected techniques beyond standard A-level, placing it moderately above average difficulty. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(G_X(t) = k(1 + 3t + 4t^2)\), \(k(1+3+4) = 1\), \(k = \frac{1}{8}\) | B1 | Working required. |
| \(G'_X(t) = k(3 + 8t)\) | M1 | Or \(\sum px = 0 \times \frac{1}{8} + 1 \times \frac{3}{8} + 2 \times \frac{4}{8} \left[= \frac{11}{8}\right]\) |
| \(E(X) = G'_X(1) = \frac{11}{8}\) | A1 | AG. Evidence of using \(t=1\) or \(\sum px\) required. CWO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(G_Z(t) = \frac{1}{8}(1 + 3t + 4t^2) \times \frac{1}{3}t^2(1 + 2t)\) | M1 | Multiply the two PGFs to obtain a single polynomial of degree 5. |
| \(\frac{1}{24}(t^2 + 5t^3 + 10t^4 + 8t^5)\) | A1 | May have \(t^2\) as a factor. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(G'_Z(t) = \frac{1}{24}(2t + 15t^2 + 40t^3 + 40t^4)\); \(G''_Z(t) = \frac{1}{24}(2 + 30t + 120t^2 + 160t^3)\) | M1 | Differentiate twice, allow one slip. |
| \(\text{Var}(X) = G''_Z(1) + G'_Z(1) - (G'_Z(1))^2 = \frac{1}{24}(312) + \frac{97}{24} - \left(\frac{97}{24}\right)^2\) | M1 | Use correct formula. |
| \(0.707\) | A1 | \(\frac{407}{576}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((Z =)\ 4\) | B1 FT | FT their final polynomial in part (b) |
## Question 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $G_X(t) = k(1 + 3t + 4t^2)$, $k(1+3+4) = 1$, $k = \frac{1}{8}$ | B1 | Working required. |
| $G'_X(t) = k(3 + 8t)$ | M1 | Or $\sum px = 0 \times \frac{1}{8} + 1 \times \frac{3}{8} + 2 \times \frac{4}{8} \left[= \frac{11}{8}\right]$ |
| $E(X) = G'_X(1) = \frac{11}{8}$ | A1 | AG. Evidence of using $t=1$ or $\sum px$ required. CWO |
---
## Question 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $G_Z(t) = \frac{1}{8}(1 + 3t + 4t^2) \times \frac{1}{3}t^2(1 + 2t)$ | M1 | Multiply the two PGFs to obtain a single polynomial of degree 5. |
| $\frac{1}{24}(t^2 + 5t^3 + 10t^4 + 8t^5)$ | A1 | May have $t^2$ as a factor. |
---
## Question 5(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $G'_Z(t) = \frac{1}{24}(2t + 15t^2 + 40t^3 + 40t^4)$; $G''_Z(t) = \frac{1}{24}(2 + 30t + 120t^2 + 160t^3)$ | M1 | Differentiate twice, allow one slip. |
| $\text{Var}(X) = G''_Z(1) + G'_Z(1) - (G'_Z(1))^2 = \frac{1}{24}(312) + \frac{97}{24} - \left(\frac{97}{24}\right)^2$ | M1 | Use correct formula. |
| $0.707$ | A1 | $\frac{407}{576}$ |
---
## Question 5(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| $(Z =)\ 4$ | B1 FT | FT their final polynomial in part (b) |5 The random variable $X$ has probability generating function $\mathrm { G } _ { X } ( \mathrm { t } )$ given by
$$\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } ) = \mathrm { k } \left( 1 + 3 \mathrm { t } + 4 \mathrm { t } ^ { 2 } \right)$$
where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Show that $\mathrm { E } ( X ) = \frac { 11 } { 8 }$.\\
The random variable $Y$ has probability generating function $\mathrm { G } _ { \gamma } ( \mathrm { t } )$ given by
$$G _ { \gamma } ( t ) = \frac { 1 } { 3 } t ^ { 2 } ( 1 + 2 t )$$
The random variables $X$ and $Y$ are independent and $\mathrm { Z } = \mathrm { X } + \mathrm { Y }$.
\item Find the probability generating function of $Z$, expressing your answer as a polynomial in $t$.
\item Use your answer to part (b) to find the value of $\operatorname { Var } ( Z )$.
\item Write down the most probable value of $Z$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2023 Q5 [9]}}