| Exam Board | Edexcel |
|---|---|
| Module | FS1 (Further Statistics 1) |
| Year | 2024 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Generating Functions |
| Type | Use PGF to find mean and variance |
| Difficulty | Challenging +1.2 This is a standard Further Statistics 1 PGF question requiring routine application of formulas (G'(1) for mean, G''(1) for variance), series expansion for probabilities, and recognition that sum of independent variables multiplies PGFs. While it involves multiple parts and Further Maths content, each step follows textbook procedures without requiring novel insight or complex problem-solving. |
| 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) | \([G_X(t) = (4 - 3t)^{-\frac{1}{2}} \Rightarrow] G_X'(t) = \frac{3}{2}(4 - 3t)^{-\frac{5}{2}}\); \([\text{So } E(X) =] G_X'(t) = \frac{3}{2}\) | M1; A1 |
| \(G_X''(t) = \frac{27}{4}(4 - 3t)^{-\frac{5}{2}}\); so \(G_X''(1) = \frac{27}{4}\) | M1; A1 Hft | 2.1; 1.1b |
| \(\text{Var}(X) = \text{" } \frac{27}{4} \text{" } + \frac{3}{2} \text{" } - \left(\frac{3}{2}\right)^2; = \mathbf{6}\) | M1; A1 | 1.1b; 1.1b |
| (b) Using Maclaurin: | M1 | 2.1 |
| \(\frac{G_X''(0)}{2!} = \left(\frac{1}{2}\right)^n \times \frac{27}{4} \times \frac{1}{32} = \left[\frac{27}{256}\right]\) | ||
| \([P(X = 2)] = \frac{1}{2} \times \frac{27}{4} \times \frac{1}{32}\left[= \frac{27}{256}\right]\) | ||
| \([P(X = 0) + P(X = 1)] = \text{" } \frac{3}{2} \times \frac{1}{8} \times \frac{1}{2}\) | A1 | 1.1b |
| \(P(X \leq 2) = \frac{203}{256}\) | A1 | 1.1b |
| Using Binomial: | M1 | 2.1 |
| \(\left[G_X(t) = \frac{1}{2}\left(1 - \frac{3}{4}\right)^{-\frac{1}{2}}\right]\) | ||
| \(\left[\frac{1}{2}\left(\ldots + \frac{\left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right)}{2}\left(-\frac{3}{4}\right)^2 + \ldots\right)\right]\left[\ldots + \frac{27}{256}t^2 + \ldots\right]\) | A1 | 1.1b |
| \(\frac{1}{2}\left(1 + \frac{3}{8}t + \frac{\left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right)}{2}\left(-\frac{3}{4}\right)^2 + \ldots\right)\) | M1 | 2.1 |
| \(P(X \leq 2) = \frac{203}{256}\) | A1 | 1.1b |
| (c) | \(\left[G_W(t) = G_{X_1}(t) \times G_{X_2}(t) \times t\right] = \frac{1}{\sqrt{4-3t}} \times \frac{1}{\sqrt{4-3t}} \times t\); \(G_Y(t) = \frac{t}{4-3t}\) | M1; A1 |
| \(G_Y(t) = \frac{1}{4} - \frac{3}{3} \text{ or } G_Y(t) = \frac{1}{4} - \frac{1}{3}\); \(Y \sim \text{Geo}\left(\frac{1}{4}\right)\) | M1; A1 | 2.1; 3.2a/2.2a |
| (d) | \(P(Y > 6) = \left(1 - \frac{1}{4}\right)^6; = \frac{729}{4096} = 0.177978\ldots\) awrt \(\mathbf{0.178}\) | M1; A1 |
(a) | $[G_X(t) = (4 - 3t)^{-\frac{1}{2}} \Rightarrow] G_X'(t) = \frac{3}{2}(4 - 3t)^{-\frac{5}{2}}$; $[\text{So } E(X) =] G_X'(t) = \frac{3}{2}$ | M1; A1 | 2.1; 1.1b |
| $G_X''(t) = \frac{27}{4}(4 - 3t)^{-\frac{5}{2}}$; so $G_X''(1) = \frac{27}{4}$ | M1; A1 Hft | 2.1; 1.1b |
| $\text{Var}(X) = \text{" } \frac{27}{4} \text{" } + \frac{3}{2} \text{" } - \left(\frac{3}{2}\right)^2; = \mathbf{6}$ | M1; A1 | 1.1b; 1.1b |
(b) **Using Maclaurin:** | M1 | 2.1 |
| $\frac{G_X''(0)}{2!} = \left(\frac{1}{2}\right)^n \times \frac{27}{4} \times \frac{1}{32} = \left[\frac{27}{256}\right]$ | | |
| $[P(X = 2)] = \frac{1}{2} \times \frac{27}{4} \times \frac{1}{32}\left[= \frac{27}{256}\right]$ | | |
| $[P(X = 0) + P(X = 1)] = \text{" } \frac{3}{2} \times \frac{1}{8} \times \frac{1}{2}$ | A1 | 1.1b |
| $P(X \leq 2) = \frac{203}{256}$ | A1 | 1.1b |
| **Using Binomial:** | M1 | 2.1 |
| $\left[G_X(t) = \frac{1}{2}\left(1 - \frac{3}{4}\right)^{-\frac{1}{2}}\right]$ | | |
| $\left[\frac{1}{2}\left(\ldots + \frac{\left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right)}{2}\left(-\frac{3}{4}\right)^2 + \ldots\right)\right]\left[\ldots + \frac{27}{256}t^2 + \ldots\right]$ | A1 | 1.1b |
| $\frac{1}{2}\left(1 + \frac{3}{8}t + \frac{\left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right)}{2}\left(-\frac{3}{4}\right)^2 + \ldots\right)$ | M1 | 2.1 |
| $P(X \leq 2) = \frac{203}{256}$ | A1 | 1.1b |
(c) | $\left[G_W(t) = G_{X_1}(t) \times G_{X_2}(t) \times t\right] = \frac{1}{\sqrt{4-3t}} \times \frac{1}{\sqrt{4-3t}} \times t$; $G_Y(t) = \frac{t}{4-3t}$ | M1; A1 | 3.1a/2.1; 1.1b |
| $G_Y(t) = \frac{1}{4} - \frac{3}{3} \text{ or } G_Y(t) = \frac{1}{4} - \frac{1}{3}$; $Y \sim \text{Geo}\left(\frac{1}{4}\right)$ | M1; A1 | 2.1; 3.2a/2.2a |
(d) | $P(Y > 6) = \left(1 - \frac{1}{4}\right)^6; = \frac{729}{4096} = 0.177978\ldots$ awrt $\mathbf{0.178}$ | M1; A1 | 3.4, 1.1b |
**Notes:**
- (a) 1st M1 for attempt to differentiate leading to $k(4-3t)^{-1.5}$; 1st A1 for $E(X) = 1.5$ or exact equivalent
- (a) 2nd M1 for attempting to differentiate again leading to $m(4-3t)^{-2.5}$
- (a) 2nd A1ft for $\frac{27}{4}$ or correct ft from their $k$ provided both Ms are scored
- (a) 3rd M1 for a correct method for finding Var$(X)$; can ft their $\frac{3}{2}$ and their $\frac{27}{4}$
- (a) 3rd A1 for 6
- (b) 1st M1 for Maclaurin to find $P(X = 2)$ condone "$\frac{27}{4} \times \frac{1}{32} [= \frac{27}{128}$ or $0.2109375]$ or putting in the form $a(1 - 0.75t)^{-0.5}$
- (b) or putting in the form $a(1 - 0.75t)^{-0.5}$
- (b) 1st A1 for a correct unsimplified prob for $P(X = 2)$ (may be in binomial expansion) Allow 0.105(468…)
- (b) 2nd M1 for use of pgf to find $P(X = 1)$ and $P(X = 0)$ or attempt 1st 3 terms of bin expansion
- (b) 2nd A1 for $\frac{203}{256}$ or exact equivalent.
- (c) 1st M1 for using product of pgf or multiplication by $t$
- (c) 1st A1 for correct unsimplified form of pgf
- (c) 2nd dM1 (dep. on 1st M1) for attempting to convert pgf to form given in the formula book or for stating Geometric alongside a correct PGF for $Y$ (may be unsimplified)
- (c) or for stating Geometric alongside a correct PGF for $Y$ (may be unsimplified)
- (c) 2nd A1 for correctly deducing the distribution of $Y$ as geometric with $p = 0.25$ [may be seen in (d)]
- (d) M1 for attempting to use geometric formula or their pgf (using correct coefficients); A1 for awrt 0.178
- **NB: Final A1 dependent on previous marks being awarded.**
- **NB: Send responses where marks are being awarded.**
---\begin{enumerate}
\item The random variable $X$ has probability generating function $\mathrm { G } _ { X } ( t )$ where
\end{enumerate}
$$\mathrm { G } _ { X } ( t ) = \frac { 1 } { \sqrt { 4 - 3 t } }$$
(a) Use calculus to find $\operatorname { Var } ( X )$
Show your working clearly.\\
(b) Find the exact value of $\mathrm { P } ( X \leqslant 2 )$
The independent random variables $X _ { 1 }$ and $X _ { 2 }$ each have the same distribution as $X$ The random variable $Y = X _ { 1 } + X _ { 2 } + 1$\\
(c) By finding the probability generating function of $Y$, state the name of the distribution of $Y$\\
(d) Hence, or otherwise, find $\mathrm { P } \left( X _ { 1 } + X _ { 2 } > 5 \right)$
\hfill \mbox{\textit{Edexcel FS1 2024 Q6 [16]}}