| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2024 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Find parameter from expectation |
| Difficulty | Standard +0.3 This is a standard S2 probability density function question requiring routine integration and algebraic manipulation. Part (b) uses the pdf property ∫f(x)dx=1, part (c) applies E(X)=∫xf(x)dx with given value, and part (d) finds the median using cumulative distribution. All techniques are textbook exercises with straightforward calculus of 1/x² and no novel problem-solving required. Slightly easier than average due to the guided structure and standard methods. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Min and max times [to complete challenge] | B1 | In context (e.g. min and max x scores B0) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\int_{a}^{b} \frac{1}{x^2} \, dx = 1\) | M1 | Attempt to integrate \(f(x)\) and \(= 1\), ignore limits |
| \(\left[-\frac{1}{x}\right]_{a}^{b} = 1 \Rightarrow -\frac{1}{b} + \frac{1}{a} = 1\) | A1 | For correct equation using correct limits into correct integration and \(= 1\) |
| \(-a + b = ab\) or \(b = a(b+1)\) | A1 | Convincingly obtained. No errors seen. \(OE \Rightarrow a = \frac{b}{b+1}\) AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(E(X) = \int_{a}^{b} \frac{1}{x} \, dx\) | M1 | Attempt to integrate \(xf(x)\). Limits \(a\) and \(b\) or \(b/(b+1)\) and \(b\) (condone \(a\) and 2 for M1). See SC for use of limits 2/3 and 2 |
| \(= \ln b - \ln a\) or \(\ln b - \ln\left(\frac{b}{b+1}\right)\) | A1 | Correct integration and limits substituted. Condone \(\ln 2 - \ln a\) |
| \(= \ln b - (\ln b - \ln(b+1)) = \ln b - \ln\!\left(\frac{b}{b+1}\right) = \ln 3\), so \(b+1 = 3\) or \(b^2 + b = 3b\) or \(\frac{b}{b+1} = 3\) | A1 | For correct equation in \(b\) only (i.e. using part (b)) |
| \(b = 2\) (AG) \(a = \frac{2}{3}\) | A1 | Both obtained correctly. Note: if \(b=2\) not shown but used can score M1 A1, A1/A0 depending on where \(b=2\) is introduced, A0. SC verification using \(b=2\) and \(a=2/3\) then integrating \(xf(x)\) from \(2/3\) to \(2\) scores M1 A1 for integration and limits substituted, then A1 for showing \(= \ln 3\). Final A0 (as verified not shown) max \(\frac{3}{4}\). |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\int_{\frac{2}{3}}^{m} \frac{1}{x^2} \, dx = 0.5\) or \(\int_{m}^{2} \frac{1}{x^2} \, dx = 0.5\) | M1 | Attempt to integrate \(f(x)\) equated to 0.5 and correct limits stated |
| \(\left[-\frac{1}{x}\right]_{\frac{2}{3}}^{m} = 0.5\) or \(\left[-\frac{1}{x}\right]_{m}^{2} = 0.5\) | A1FT | Correct integration FT their \(a\) |
| \(\left[-\frac{1}{m} + \frac{3}{2} = 0.5\right]\) or \(\left[-\frac{1}{2} + \frac{1}{m} = 0.5\right]\); \(m = 1\) | A1 |
## Question 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Min and max times [to complete challenge] | B1 | In context (e.g. min and max x scores **B0**) |
## Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\int_{a}^{b} \frac{1}{x^2} \, dx = 1$ | M1 | Attempt to integrate $f(x)$ and $= 1$, ignore limits |
| $\left[-\frac{1}{x}\right]_{a}^{b} = 1 \Rightarrow -\frac{1}{b} + \frac{1}{a} = 1$ | A1 | For correct equation using correct limits into correct integration and $= 1$ |
| $-a + b = ab$ or $b = a(b+1)$ | A1 | Convincingly obtained. No errors seen. $OE \Rightarrow a = \frac{b}{b+1}$ **AG** |
## Question 6(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $E(X) = \int_{a}^{b} \frac{1}{x} \, dx$ | M1 | Attempt to integrate $xf(x)$. Limits $a$ and $b$ or $b/(b+1)$ and $b$ (condone $a$ and 2 for M1). See SC for use of limits 2/3 and 2 |
| $= \ln b - \ln a$ or $\ln b - \ln\left(\frac{b}{b+1}\right)$ | A1 | Correct integration and limits substituted. Condone $\ln 2 - \ln a$ |
| $= \ln b - (\ln b - \ln(b+1)) = \ln b - \ln\!\left(\frac{b}{b+1}\right) = \ln 3$, so $b+1 = 3$ or $b^2 + b = 3b$ or $\frac{b}{b+1} = 3$ | A1 | For correct equation in $b$ only (i.e. using part (b)) |
| $b = 2$ **(AG)** $a = \frac{2}{3}$ | A1 | Both obtained correctly. Note: if $b=2$ not shown but used can score **M1 A1, A1/A0** depending on where $b=2$ is introduced, **A0**. SC verification using $b=2$ and $a=2/3$ then integrating $xf(x)$ from $2/3$ to $2$ scores **M1 A1** for integration and limits substituted, then **A1** for showing $= \ln 3$. Final **A0** (as verified not shown) max $\frac{3}{4}$. |
## Question 6(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\int_{\frac{2}{3}}^{m} \frac{1}{x^2} \, dx = 0.5$ or $\int_{m}^{2} \frac{1}{x^2} \, dx = 0.5$ | M1 | Attempt to integrate $f(x)$ equated to 0.5 and correct limits stated |
| $\left[-\frac{1}{x}\right]_{\frac{2}{3}}^{m} = 0.5$ or $\left[-\frac{1}{x}\right]_{m}^{2} = 0.5$ | A1FT | Correct integration FT their $a$ |
| $\left[-\frac{1}{m} + \frac{3}{2} = 0.5\right]$ or $\left[-\frac{1}{2} + \frac{1}{m} = 0.5\right]$; $m = 1$ | A1 | |6 The time, $X$ hours, taken by a large number of people to complete a challenge is modelled by the probability density function given by
$$f ( x ) = \left\{ \begin{array} { c l }
\frac { 1 } { x ^ { 2 } } & a \leqslant x \leqslant b \\
0 & \text { otherwise }
\end{array} \right.$$
where $a$ and $b$ are constants.
\begin{enumerate}[label=(\alph*)]
\item State what the constants $a$ and $b$ represent in this context.
\item Show that $a = \frac { b } { b + 1 }$.\\
It is given that $\mathrm { E } ( X ) = \ln 3$.
\item Show that $b = 2$ and find the value of $a$.\\
\includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-09_2726_35_97_20}
\item Find the median of $X$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2024 Q6 [11]}}