| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2021 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Generating Functions |
| Type | Given PGF manipulation and properties |
| Difficulty | Standard +0.3 This is a straightforward application of standard PGF techniques from Further Maths Statistics. Part (a) uses basic probability axioms, (b) is direct PGF construction, (c) applies the independence property G_Z(t) = G_X(t)G_Y(t), and (d) uses standard PGF formulas for variance. All steps are routine textbook exercises with no novel insight required, making it slightly easier than average for A-level Further Maths. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks |
|---|---|
| \((1+4+9+16)k = 1\) so \(k = \frac{1}{30}\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(G_X(t) = \frac{1}{30}t + \frac{4}{30}t^2 + \frac{9}{30}t^3 + \frac{16}{30}t^4\) | M1A1 | Using their \(k\) in a polynomial, at least two terms correct for their \(k\). |
| Answer | Marks | Guidance |
|---|---|---|
| \(\left(\frac{1}{30}t+\frac{4}{30}t^2+\frac{9}{30}t^3+\frac{16}{30}t^4\right)\left(\frac{1}{4}+\frac{1}{2}t+\frac{1}{4}t^2\right)\) | M1 | Method and attempt to multiply. |
| \(\frac{1}{120}\left(t+6t^2+18t^3+38t^4+41t^5+16t^6\right)\) | M1A1 | Multiplication to obtain single polynomial of order 6. |
| Answer | Marks | Guidance |
|---|---|---|
| \(G''(t) = \frac{1}{120}(12+108t+456t^2+820t^3+480t^4)\) | M1 | Differentiate twice. |
| \(\text{Var}(X) = G''(1) + \frac{13}{3} - \left(\frac{13}{3}\right)^2\) | M1 | Use correct formula. |
| \(\frac{1876}{120}+\frac{13}{3}-\frac{169}{9} = \frac{107}{90}\) or \(1.19\) | A1 | CAO |
## Question 5(a):
$(1+4+9+16)k = 1$ so $k = \frac{1}{30}$ | B1 |
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## Question 5(b):
$G_X(t) = \frac{1}{30}t + \frac{4}{30}t^2 + \frac{9}{30}t^3 + \frac{16}{30}t^4$ | M1A1 | Using their $k$ in a polynomial, at least two terms correct for their $k$.
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## Question 5(c):
$\left(\frac{1}{30}t+\frac{4}{30}t^2+\frac{9}{30}t^3+\frac{16}{30}t^4\right)\left(\frac{1}{4}+\frac{1}{2}t+\frac{1}{4}t^2\right)$ | M1 | Method and attempt to multiply.
$\frac{1}{120}\left(t+6t^2+18t^3+38t^4+41t^5+16t^6\right)$ | M1A1 | Multiplication to obtain single polynomial of order 6.
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## Question 5(d):
Given: $G'(1) = \frac{13}{3}$
$G''(t) = \frac{1}{120}(12+108t+456t^2+820t^3+480t^4)$ | M1 | Differentiate twice.
$\text{Var}(X) = G''(1) + \frac{13}{3} - \left(\frac{13}{3}\right)^2$ | M1 | Use correct formula.
$\frac{1876}{120}+\frac{13}{3}-\frac{169}{9} = \frac{107}{90}$ or $1.19$ | A1 | CAO5 The random variable $X$ is such that $\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = \mathrm { kr } ^ { 2 }$ for $r = 1,2,3,4$, where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $k$.
\item Find the probability generating function $\mathrm { G } _ { X } ( \mathrm { t } )$ of $X$.\\
The random variable $Y$ has probability generating function $\mathrm { G } _ { Y } ( \mathrm { t } ) = \frac { 1 } { 4 } + \frac { 1 } { 2 } \mathrm { t } + \frac { 1 } { 4 } \mathrm { t } ^ { 2 }$.\\
The random variable $Z$ is the sum of $X$ and $Y$.
\item Assuming that $X$ and $Y$ are independent, find the probability generating function $\mathrm { G } _ { \mathrm { Z } } ( \mathrm { t } )$ of $Z$ as a polynomial in $t$.
\item Given that $\mathrm { E } ( \mathrm { Z } ) = \frac { 13 } { 3 }$, use $\mathrm { G } _ { \mathrm { Z } } ( \mathrm { t } )$ to find $\operatorname { Var } ( \mathrm { Z } )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2021 Q5 [9]}}