| Exam Board | Edexcel |
|---|---|
| Module | FS1 (Further Statistics 1) |
| Year | 2023 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Generating Functions |
| Type | Identify distribution from PGF |
| Difficulty | Challenging +1.2 This is a Further Maths Statistics question requiring recognition of negative binomial PGF form, standard calculus techniques for expectation/variance, and careful manipulation for part (d). While it requires multiple techniques and is inherently harder than A-level content, the steps are mostly procedural once the distribution is identified. The final part requires some care with index manipulation but follows standard PGF methods. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| (a) NegBin\((r, p)\) has pgf \(\left[\frac{pt}{1-(1-p)t}\right]^r\) and identify the connection \(\mathbf{\text{NegBin}(2, \frac{1}{3})}\) | M1, A1 | 2.1, 2.2a |
| (b) e.g. no. of rolls to achieve 5 or 6 (so that \(p = \frac{1}{3}\)) twice (oe) | B1 ft | 3.3 |
| (c)(i) \(G_X'(t) = \frac{2t(3-2t)^2 - (-2) \times 2(3-2t)t^2}{(3-2t)^3}\) or \(\frac{6t}{(3-2t)^3}\) | M1, A1 | 2.1, 1.1b |
| \(E(Y) = G_X'(1) = \mathbf{6}\) | A1 | 1.1b |
| (ii) \(G_X''(t) = \frac{6(3-2t)^3 - (-2) \times 3(3-2t)^2 \times 6t}{(3-2t)^6}\) or \(\frac{18 + 24t}{(3-2t)^4}\) | M1 | 2.1 |
| \(G_X''(1) = \mathbf{42}\) | A1 | 1.1b |
| Var\((X) = ''42'' + ''6'' - ''6''^2 = \mathbf{12}\) | M1, A1 | 1.1b, 1.1b |
| (d) \(G_Y(t) = t^{10} \times \frac{1}{9}\left[1 - \frac{2}{3}t^3\right]^{-2} = \frac{1}{9}\left[1 + \ldots \frac{(-2)(-3)(-4)}{3!}\left(\frac{2}{3}\right)^3 t^9 \ldots\right]\) | M1 A1 | 2.1, 1.1b |
| \(\text{P}(Y = 19) = \frac{32}{243}\) | A1 | 1.1b |
| ALT Identify that \(Y = 3X + 4\) | M1 | |
| \((Y = 19 \text{ requires } X = 5 \text{ so })\) \(\text{P}(X = 5) = \binom{4}{1}\left(\frac{1}{3}\right)\left(\frac{2}{3}\right)^3\left(\frac{1}{3}\right)\) | A1 |
**(a)** NegBin$(r, p)$ has pgf $\left[\frac{pt}{1-(1-p)t}\right]^r$ and identify the connection $\mathbf{\text{NegBin}(2, \frac{1}{3})}$ | M1, A1 | 2.1, 2.2a | (2)
**(b)** e.g. no. of rolls to achieve 5 or 6 (so that $p = \frac{1}{3}$) twice (oe) | B1 ft | 3.3 | (1)
**(c)(i)** $G_X'(t) = \frac{2t(3-2t)^2 - (-2) \times 2(3-2t)t^2}{(3-2t)^3}$ or $\frac{6t}{(3-2t)^3}$ | M1, A1 | 2.1, 1.1b |
$E(Y) = G_X'(1) = \mathbf{6}$ | A1 | 1.1b |
**(ii)** $G_X''(t) = \frac{6(3-2t)^3 - (-2) \times 3(3-2t)^2 \times 6t}{(3-2t)^6}$ or $\frac{18 + 24t}{(3-2t)^4}$ | M1 | 2.1 |
$G_X''(1) = \mathbf{42}$ | A1 | 1.1b |
Var$(X) = ''42'' + ''6'' - ''6''^2 = \mathbf{12}$ | M1, A1 | 1.1b, 1.1b | (7)
**(d)** $G_Y(t) = t^{10} \times \frac{1}{9}\left[1 - \frac{2}{3}t^3\right]^{-2} = \frac{1}{9}\left[1 + \ldots \frac{(-2)(-3)(-4)}{3!}\left(\frac{2}{3}\right)^3 t^9 \ldots\right]$ | M1 A1 | 2.1, 1.1b |
$\text{P}(Y = 19) = \frac{32}{243}$ | A1 | 1.1b | (3)
**ALT** Identify that $Y = 3X + 4$ | M1 |
$(Y = 19 \text{ requires } X = 5 \text{ so })$ $\text{P}(X = 5) = \binom{4}{1}\left(\frac{1}{3}\right)\left(\frac{2}{3}\right)^3\left(\frac{1}{3}\right)$ | A1 | | (13)
**Notes:**
(a) M1 for identifying the NegBin distribution (allow NB for NegBin)
A1 for $r = 2$ and $p = \frac{1}{3}$
(b) B1 ft for identifying a suitable definition for $X$ using a (fair) die, with $p = \frac{1}{3}$ and the second occurrence of the event, only ft their NegBin distribution in (a). A finite number of rolls is B0
(c)(i) 1st M1 for attempt to differentiate quotient or product. At least one $uv'$ style term correct. 1st A1 for a fully correct first derivative (needn't be simplified)
2nd A1 for E$(X) = 6$ NB this A1 depends on M1 only but M1A0A1 is possible
(ii) 2nd M1 for attempt to diff' quotient or product again. At least one $uv'$ style term correct.
3rd A1 for 42 (may be given for incorrect G'' provided their G''$(1)$ gives 42 and M1 scored)
Note all powers of $(3 - 2t)$ equal 1 when $t = 1$ is substituted so can be used as a check
3rd M1 for correct use of pgf to find Var$(X)$
4th A1 dep on M3 for 12
(d) M1 for writing pgf in suitable form to carry out binomial expansion
1st A1 for a correct expression for coefficient of $t^{19}$
2nd A1 for $\frac{32}{243}$ or exact equivalent
**ALT** M1 for identifying connection $Y = 3X + 4$
1st A1 for a correct numerical probability expression for P$(X = 5)$
**(Total: 13 marks)**
---\begin{enumerate}
\item The discrete random variable $X$ has probability generating function
\end{enumerate}
$$\mathrm { G } _ { X } ( t ) = \frac { t ^ { 2 } } { ( 3 - 2 t ) ^ { 2 } }$$
(a) Specify the distribution of $X$
A fair die is rolled repeatedly.\\
(b) Describe an outcome that could be modelled by the random variable $X$\\
(c) Use calculus and $\mathrm { G } _ { X } ( t )$ to find\\
(i) $\mathrm { E } ( X )$\\
(ii) $\operatorname { Var } ( X )$
The discrete random variable $Y$ has probability generating function
$$\mathrm { G } _ { Y } ( t ) = \frac { t ^ { 10 } } { \left( 3 - 2 t ^ { 3 } \right) ^ { 2 } }$$
(d) Find the exact value of $\mathrm { P } ( Y = 19 )$
\hfill \mbox{\textit{Edexcel FS1 2023 Q6 [13]}}