| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2018 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Single-piece PDF with k |
| Difficulty | Moderate -0.3 This is a straightforward continuous probability distribution question requiring standard techniques: finding k by integration (equals 1), computing E(X) using the definition, and solving probability equations. All parts follow routine procedures with no novel problem-solving required, making it slightly easier than average for A-level statistics. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int_5^{10} \frac{k}{x^2} dx = 1\) | M1 | Attempt integration f(x) and '= 1'; ignore limits |
| \(\left[-\frac{k}{x}\right]_5^{10} = 1\) oe, \(\left(\frac{k}{5} - \frac{k}{10} = 1\right)\) | A1 | Correct integration and limits and '= 1' |
| \(k = 10\) | A1 | No errors seen |
| Answer | Marks | Guidance |
|---|---|---|
| \(10\int_5^{10} \frac{1}{x} dx\) | M1 | Attempt integ \(xf(x)\); ignore limits |
| \(10\left[\ln x\right]_5^{10}\), or \(10(\ln 10 - \ln 5)\) | ||
| \(= 10\ln 2\) | A1 | No errors seen |
| Answer | Marks | Guidance |
|---|---|---|
| \(10\int_9^{10} \frac{1}{x^2} dx\), \(\left(10\left[-\frac{1}{x}\right]_9^{10}\right)\) | M1 | Attempt integ f(x) with correct limits |
| \(10\left[-\frac{1}{10} + \frac{1}{9}\right]\) | A1 | Substitute correct limits in correct integration |
| \(= \frac{1}{9}\) or \(0.111\) (3 sf) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int_5^a \frac{k}{x^2} dx = 0.6\), \(10\left[-\frac{1}{x}\right]_5^a = 0.6\) | M1 | Attempt integration of f(x) with correct limits and = 0.6 |
| \(10\left[\frac{1}{5} - \frac{1}{a}\right] = 0.6\) | A1 | Substitute correct limits in correct integration |
| \(a = \frac{50}{7}\) or \(7.14\) (3 sf) | A1 |
## Question 6(i):
$\int_5^{10} \frac{k}{x^2} dx = 1$ | M1 | Attempt integration f(x) and '= 1'; ignore limits
$\left[-\frac{k}{x}\right]_5^{10} = 1$ oe, $\left(\frac{k}{5} - \frac{k}{10} = 1\right)$ | A1 | Correct integration and limits and '= 1'
$k = 10$ | A1 | No errors seen
**Total: 3 marks**
---
## Question 6(ii):
$10\int_5^{10} \frac{1}{x} dx$ | M1 | Attempt integ $xf(x)$; ignore limits
$10\left[\ln x\right]_5^{10}$, or $10(\ln 10 - \ln 5)$ | |
$= 10\ln 2$ | A1 | No errors seen
**Total: 2 marks**
---
## Question 6(iii):
$10\int_9^{10} \frac{1}{x^2} dx$, $\left(10\left[-\frac{1}{x}\right]_9^{10}\right)$ | M1 | Attempt integ f(x) with correct limits
$10\left[-\frac{1}{10} + \frac{1}{9}\right]$ | A1 | Substitute correct limits in correct integration
$= \frac{1}{9}$ or $0.111$ (3 sf) | A1 |
**Total: 3 marks**
---
## Question 6(iv):
$\int_5^a \frac{k}{x^2} dx = 0.6$, $10\left[-\frac{1}{x}\right]_5^a = 0.6$ | M1 | Attempt integration of f(x) with correct limits and = 0.6
$10\left[\frac{1}{5} - \frac{1}{a}\right] = 0.6$ | A1 | Substitute correct limits in correct integration
$a = \frac{50}{7}$ or $7.14$ (3 sf) | A1 |
**Total: 3 marks**
---6 The time, in minutes, taken by people to complete a test is modelled by the continuous random variable $X$ with probability density function given by
$$f ( x ) = \begin{cases} \frac { k } { x ^ { 2 } } & 5 \leqslant x \leqslant 10 \\ 0 & \text { otherwise } \end{cases}$$
where $k$ is a constant.\\
(i) Show that $k = 10$.\\
(ii) Show that $\mathrm { E } ( X ) = 10 \ln 2$.\\
(iii) Find $\mathrm { P } ( X > 9 )$.\\
(iv) Given that $\mathrm { P } ( X < a ) = 0.6$, find $a$.\\
\hfill \mbox{\textit{CAIE S2 2018 Q6 [11]}}