| Exam Board | Edexcel |
|---|---|
| Module | FS2 (Further Statistics 2) |
| Session | Specimen |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Cumulative distribution functions |
| Type | Distribution of order statistics |
| Difficulty | Standard +0.8 This is a Further Maths Statistics question on order statistics requiring multiple techniques: finding variance from a CDF via integration, identifying the mode from a pdf, analyzing skewness, and solving a probability equation. While each part uses standard methods, the combination of techniques and the need to work with order statistics (maximum of three variables) places it moderately above average difficulty for A-level. |
| Spec | 5.02a Discrete probability distributions: general5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{2}{5}\) o.e. | B1 | May be implied by a correct answer |
| \(\text{"their } \left(\frac{2}{5}\right)^3\text{"}\) | M1 | May be implied by a correct answer |
| \(\frac{8}{125}\) o.e. | A1 |
| Answer | Marks |
|---|---|
| Realising that firstly need to find pdf \(f(y)\) and attempt to differentiate \(F(y)\) | M1 |
| Continuing the argument with an attempt to integrate \(y \times\) "their \(f(y)\)" \(y^n \to y^{n+1}\) | M1 |
| Integrating \(y^2 \times\) "their \(f(y)\)" \(-\) ["their \(E(Y)\)"]\(^2\) \(y^n \to y^{n+1}\) | M1 |
| Complete correct solution no errors | A1* |
| Answer | Marks |
|---|---|
| 5 only | B1 |
| Explain their reason by either an accurate sketch or \(\frac{df(y)}{dy} > 0\) therefore an increasing function o.e. | B1 |
| Answer | Marks |
|---|---|
| Explaining the reason for their answer. Follow through their part(b) or mean from(d) and mode from(c). A correct sketch of "their \(f(y)\)" – may be seen anywhere in question or ft their mean and mode plus a correct conclusion | B1ft |
| NB: Watch for gaming. A student who writes both negative skew with a reason and positive skew with a reason. Please send these to your Team Leader |
| Answer | Marks |
|---|---|
| Attempting to translate the problem into an equation using \(2k\) and \(k\). Allow if the brackets are missing e.g. \(\frac{2k^3}{125} - \frac{k^3}{125}\). No need for the 0.189 | M1 |
| A correct equation in any form | A1 |
| A correct answer only | A1 |
**(a)**
| $\frac{2}{5}$ o.e. | B1 | May be implied by a correct answer |
| $\text{"their } \left(\frac{2}{5}\right)^3\text{"}$ | M1 | May be implied by a correct answer |
| $\frac{8}{125}$ o.e. | A1 | |
**(b)**
| Realising that firstly need to find pdf $f(y)$ and attempt to differentiate $F(y)$ | M1 | |
| Continuing the argument with an attempt to integrate $y \times$ "their $f(y)$" $y^n \to y^{n+1}$ | M1 | |
| Integrating $y^2 \times$ "their $f(y)$" $-$ ["their $E(Y)$"]$^2$ $y^n \to y^{n+1}$ | M1 | |
| Complete correct solution no errors | A1* | |
**(c)**
| 5 only | B1 | |
| Explain their reason by either an accurate sketch or $\frac{df(y)}{dy} > 0$ therefore an increasing function o.e. | B1 | |
**(d)**
| Explaining the reason for their answer. Follow through their part(b) or mean from(d) and mode from(c). A correct sketch of "their $f(y)$" – may be seen anywhere in question or ft their mean and mode plus a correct conclusion | B1ft | |
| NB: Watch for gaming. A student who writes both negative skew with a reason and positive skew with a reason. Please send these to your Team Leader | | |
**(e)**
| Attempting to translate the problem into an equation using $2k$ and $k$. Allow if the brackets are missing e.g. $\frac{2k^3}{125} - \frac{k^3}{125}$. No need for the 0.189 | M1 | |
| A correct equation in any form | A1 | |
| A correct answer only | A1 | |
---\begin{enumerate}
\item The three independent random variables $A , B$ and $C$ each have a continuous uniform distribution over the interval $[ 0,5 ]$.\\
(a) Find the probability that $A , B$ and $C$ are all greater than 3
\end{enumerate}
The random variable $Y$ represents the maximum value of $A , B$ and $C$.\\
The cumulative distribution function of $Y$ is
$$\mathrm { F } ( y ) = \begin{cases} 0 & y < 0 \\ \frac { y ^ { 3 } } { 125 } & 0 \leqslant y \leqslant 5 \\ 1 & y > 5 \end{cases}$$
(b) Using algebraic integration, show that $\operatorname { Var } ( Y ) = 0.9375$\\
(c) Find the mode of $Y$, giving a reason for your answer.\\
(d) Describe the skewness of the distribution of $Y$. Give a reason for your answer.\\
(e) Find the value of $k$ such that $\mathrm { P } ( k < Y < 2 k ) = 0.189$
\hfill \mbox{\textit{Edexcel FS2 Q1 [13]}}