| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2020 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Generating Functions |
| Type | Derive standard distribution PGF |
| Difficulty | Standard +0.3 This is a standard Further Maths Statistics question requiring derivation of the binomial PGF using the binomial theorem and then applying standard PGF formulas (G'(1) and G''(1)+G'(1)-[G'(1)]²) to find mean and variance. While it involves algebraic manipulation and understanding of PGFs, it follows a well-established template with no novel insight required, making it slightly easier than average for Further Maths content. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(X=r) = {}^nC_r\, p^r(1-p)^{n-r}\) or \({}^nC_r\, p^r q^{n-r}\) | B1 | |
| \(G_X(t) = \sum_0^n nC_r\, p^r(1-p)^{n-r}\,t^r\) | M1 | Accept minimum of 4 terms including the last. Shown with specific value of \(n\) is M0 |
| \(\sum_0^n nC_r(pt)^r(1-p)^{n-r} = (q+pt)^n\) | A1 | At least one intermediate step to be shown, with \(p\) and \(t\) grouped. AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(G'_X(t) = n(q+pt)^{n-1} \times p\) | M1 | |
| So \(E(X) = G'_X(1) = np(q+p)^{n-1}\) and \(q+p=1\), so \(E(X) = np\) | A1 | |
| \(G''_X(t) = n(n-1)(q+pt)^{n-2} \times p \times p\) | M1 | |
| \(\text{Var}(X) = n(n-1)p^2 + np - (np)^2\) | M1 | |
| \(np(1-p)\) | A1 |
## Question 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(X=r) = {}^nC_r\, p^r(1-p)^{n-r}$ or ${}^nC_r\, p^r q^{n-r}$ | B1 | |
| $G_X(t) = \sum_0^n nC_r\, p^r(1-p)^{n-r}\,t^r$ | M1 | Accept minimum of 4 terms including the last. Shown with specific value of $n$ is M0 |
| $\sum_0^n nC_r(pt)^r(1-p)^{n-r} = (q+pt)^n$ | A1 | At least one intermediate step to be shown, with $p$ and $t$ grouped. AG |
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## Question 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $G'_X(t) = n(q+pt)^{n-1} \times p$ | M1 | |
| So $E(X) = G'_X(1) = np(q+p)^{n-1}$ and $q+p=1$, so $E(X) = np$ | A1 | |
| $G''_X(t) = n(n-1)(q+pt)^{n-2} \times p \times p$ | M1 | |
| $\text{Var}(X) = n(n-1)p^2 + np - (np)^2$ | M1 | |
| $np(1-p)$ | A1 | |5 The random variable $X$ has the binomial distribution $\mathrm { B } ( n , p )$.
\begin{enumerate}[label=(\alph*)]
\item Write down an expression for $\mathrm { P } ( \mathrm { X } = \mathrm { r } )$ and hence show that the probability generating function of $X$ is $( \mathrm { q } + \mathrm { pt } ) ^ { \mathrm { n } }$, where $\mathrm { q } = 1 - \mathrm { p }$.
\item Use the probability generating function of $X$ to prove that $\mathrm { E } ( \mathrm { X } ) = \mathrm { np }$ and $\operatorname { Var } ( \mathrm { X } ) = \mathrm { np } ( 1 - \mathrm { p } )$. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2020 Q5 [8]}}