| Exam Board | Edexcel |
|---|---|
| Module | FS1 (Further Statistics 1) |
| Year | 2019 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Generating Functions |
| Type | Given PGF manipulation and properties |
| Difficulty | Challenging +1.2 This is a Further Maths Statistics question on PGFs requiring standard techniques: using G(1)=1 to find k, applying G'(1) and G''(1) formulas for variance, and differentiating three times to find P(X=3). While it involves logarithmic differentiation and algebraic manipulation, these are routine applications of PGF theory without novel problem-solving or geometric insight. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(G(1) = 1 \Rightarrow k\ln 2 = 1\) so \(k = \dfrac{1}{\ln 2}\) | B1 | For finding \(k\) (must be exact) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\left\{G(t) = \dfrac{1}{\ln 2}[\ln 2 - \ln(2-t)]\right\} \Rightarrow G'(t) = \dfrac{1}{\ln 2}\left[\dfrac{1}{2-t}\right]\) or \(\dfrac{1}{\ln 2}(2-t)^{-1}\) | M1, A1 | 1st M1 for attempt to differentiate \(G(t)\); 1st A1 for correct first derivative (condone \(k\) or use of \(\frac{1}{\ln 2} = \text{awrt } 1.44\)) |
| \([\text{E}(X) =]\ G'(1) = \dfrac{1}{\ln 2}\) | A1 | 2nd A1 for correct \(E(X)\) or \(G'(1)\) (allow awrt 1.44) |
| \(G''(t) = \dfrac{1}{\ln 2} \times \left[\dfrac{1}{(2-t)^2}\right]\) | M1, A1 | 2nd M1 for attempting second derivative; 3rd A1 for correct 2nd derivative |
| \(\text{Var}(X) = G''(1) + G'(1) - [G'(1)]^2 = \dfrac{1}{\ln 2} + \dfrac{1}{\ln 2} - \left(\dfrac{1}{\ln 2}\right)^2\) | M1 | 3rd M1 for correct method for \(\text{Var}(X)\) (some substitution into correct formula) |
| \(= \dfrac{1}{\ln 2}\left(2 - \dfrac{1}{\ln 2}\right)\) | A1 | 4th A1; o.e. but must simplify i.e. collect like terms. NB 0.8040211 is A0 unless exact answer seen |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(X = 3) = \text{coefficient of } t^3\) by Maclaurin, need \(G'''(0)\) | M1 | For suitable strategy; need mention of coefficient of \(t^3\) and \([G'''(t)\) or \(G'''(0)]\) (condone \(G'''(1)\)) |
| \(G'''(t) = \dfrac{1}{\ln 2} \cdot \dfrac{2}{(2-t)^3}\) | A1ft | For 3rd derivative, ft their 2nd derivative in (b) (provided \(G''(t)\) not const). Correct \(G'''(t)\) or \(G'''(0)\) scores 1st M1 1st A1ft |
| \(P(X = 3) = \dfrac{G'''(0)}{3!}\) | M1 | For translating Maclaurin to probability (a correct expression) |
| \(= \dfrac{\frac{1}{4\ln 2}}{6} = \dfrac{1}{24\ln 2} = 0.0601122...\) awrt \(\mathbf{0.0601}\) | A1 | For \(\frac{1}{24\ln 2}\) or awrt 0.0601 |
# Question 6:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $G(1) = 1 \Rightarrow k\ln 2 = 1$ so $k = \dfrac{1}{\ln 2}$ | B1 | For finding $k$ (must be exact) |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $\left\{G(t) = \dfrac{1}{\ln 2}[\ln 2 - \ln(2-t)]\right\} \Rightarrow G'(t) = \dfrac{1}{\ln 2}\left[\dfrac{1}{2-t}\right]$ or $\dfrac{1}{\ln 2}(2-t)^{-1}$ | M1, A1 | 1st M1 for attempt to differentiate $G(t)$; 1st A1 for correct first derivative (condone $k$ or use of $\frac{1}{\ln 2} = \text{awrt } 1.44$) |
| $[\text{E}(X) =]\ G'(1) = \dfrac{1}{\ln 2}$ | A1 | 2nd A1 for correct $E(X)$ or $G'(1)$ (allow awrt 1.44) |
| $G''(t) = \dfrac{1}{\ln 2} \times \left[\dfrac{1}{(2-t)^2}\right]$ | M1, A1 | 2nd M1 for attempting second derivative; 3rd A1 for correct 2nd derivative |
| $\text{Var}(X) = G''(1) + G'(1) - [G'(1)]^2 = \dfrac{1}{\ln 2} + \dfrac{1}{\ln 2} - \left(\dfrac{1}{\ln 2}\right)^2$ | M1 | 3rd M1 for correct method for $\text{Var}(X)$ (some substitution into correct formula) |
| $= \dfrac{1}{\ln 2}\left(2 - \dfrac{1}{\ln 2}\right)$ | A1 | 4th A1; o.e. but must simplify i.e. collect like terms. NB 0.8040211 is A0 unless exact answer seen |
## Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(X = 3) = \text{coefficient of } t^3$ by Maclaurin, need $G'''(0)$ | M1 | For suitable strategy; need mention of coefficient of $t^3$ **and** $[G'''(t)$ or $G'''(0)]$ (condone $G'''(1)$) |
| $G'''(t) = \dfrac{1}{\ln 2} \cdot \dfrac{2}{(2-t)^3}$ | A1ft | For 3rd derivative, ft their 2nd derivative in (b) (provided $G''(t)$ not const). Correct $G'''(t)$ or $G'''(0)$ scores 1st M1 1st A1ft |
| $P(X = 3) = \dfrac{G'''(0)}{3!}$ | M1 | For translating Maclaurin to probability (a correct expression) |
| $= \dfrac{\frac{1}{4\ln 2}}{6} = \dfrac{1}{24\ln 2} = 0.0601122...$ awrt $\mathbf{0.0601}$ | A1 | For $\frac{1}{24\ln 2}$ or awrt 0.0601 |
---\begin{enumerate}
\item The discrete random variable $X$ has probability generating function
\end{enumerate}
$$\mathrm { G } _ { X } ( t ) = k \ln \left( \frac { 2 } { 2 - t } \right)$$
where $k$ is a constant.\\
(a) Find the exact value of $k$\\
(b) Find the exact value of $\operatorname { Var } ( X )$\\
(c) Find $\mathrm { P } ( X = 3 )$
\hfill \mbox{\textit{Edexcel FS1 2019 Q6 [12]}}