| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2019 |
| Session | November |
| Marks | 10 |
| 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 standard continuous probability distribution question requiring routine integration techniques: finding k by integrating to 1, calculating a probability by integration, and finding variance using E(X²) - [E(X)]². All steps are textbook exercises with straightforward polynomial integration, making it slightly easier than average A-level difficulty. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(k\int_0^3 (3x - x^2)\,dx = 1\) | M1 | Attempt to integrate \(f(x)\) and \(= 1\) |
| \(k\left[\frac{3}{2}x^2 - \frac{x^3}{3}\right]_0^3\); \(k\left(\frac{27}{2} - \frac{27}{3}\right) = 1\) | A1 | Correct integral and limits |
| \(k = \frac{2}{9}\) | A1 | AG. No errors seen |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{2}{9}\int_1^2 (3x - x^2)\,dx = \frac{2}{9}\left[\frac{3}{2}x^2 - \frac{x^3}{3}\right]_1^2 = \frac{2}{9}\times\left(6 - \frac{8}{3} - \frac{3}{2} + \frac{1}{3}\right)\) | M1 | Attempt to integrate \(f(x)\,dx\) with limits 1 and 2. OE |
| \(\frac{13}{27}\) or \(0.481\) (3 sf) | A1 | |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(y = 3x - x^2\) symmetrical about \(x = \frac{3}{2}\) | M1 | Attempt \(\frac{2}{9}\int_0^3 (3x^2 - x^3)\,dx\) |
| \(E(X) = \frac{3}{2}\) | A1 | |
| \(\frac{2}{9}\int_0^3 (3x^3 - x^4)\,dx\) | M1 | Attempt to integrate \(x^2 f(x)\) |
| \(= \frac{2}{9}\left[\frac{3x^4}{4} - \frac{x^5}{5}\right]_0^3 \left(= \frac{2}{9} \times \frac{243}{20} = \frac{27}{10}\right)\); \(\frac{27}{10} - \left(\frac{3}{2}\right)^2\) | M1 | Subtract their \((E(X))^2\) from their integral \(x^2 f(x)\) with correct limits substituted |
| \(\frac{9}{20}\) or \(0.45\) | A1 | |
| Total: 5 |
## Question 6(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $k\int_0^3 (3x - x^2)\,dx = 1$ | M1 | Attempt to integrate $f(x)$ and $= 1$ |
| $k\left[\frac{3}{2}x^2 - \frac{x^3}{3}\right]_0^3$; $k\left(\frac{27}{2} - \frac{27}{3}\right) = 1$ | A1 | Correct integral and limits |
| $k = \frac{2}{9}$ | A1 | AG. No errors seen |
| **Total: 3** | | |
---
## Question 6(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{2}{9}\int_1^2 (3x - x^2)\,dx = \frac{2}{9}\left[\frac{3}{2}x^2 - \frac{x^3}{3}\right]_1^2 = \frac{2}{9}\times\left(6 - \frac{8}{3} - \frac{3}{2} + \frac{1}{3}\right)$ | M1 | Attempt to integrate $f(x)\,dx$ with limits 1 and 2. OE |
| $\frac{13}{27}$ or $0.481$ (3 sf) | A1 | |
| **Total: 2** | | |
---
## Question 6(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y = 3x - x^2$ symmetrical about $x = \frac{3}{2}$ | M1 | Attempt $\frac{2}{9}\int_0^3 (3x^2 - x^3)\,dx$ |
| $E(X) = \frac{3}{2}$ | A1 | |
| $\frac{2}{9}\int_0^3 (3x^3 - x^4)\,dx$ | M1 | Attempt to integrate $x^2 f(x)$ |
| $= \frac{2}{9}\left[\frac{3x^4}{4} - \frac{x^5}{5}\right]_0^3 \left(= \frac{2}{9} \times \frac{243}{20} = \frac{27}{10}\right)$; $\frac{27}{10} - \left(\frac{3}{2}\right)^2$ | M1 | Subtract their $(E(X))^2$ from their integral $x^2 f(x)$ with correct limits substituted |
| $\frac{9}{20}$ or $0.45$ | A1 | |
| **Total: 5** | | |
---6 A random variable $X$ has probability density function given by
$$\mathrm { f } ( x ) = \begin{cases} k \left( 3 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
(i) Show that $k = \frac { 2 } { 9 }$.\\
(ii) Find $\mathrm { P } ( 1 \leqslant X \leqslant 2 )$.\\
(iii) Find $\operatorname { Var } ( X )$.\\
\hfill \mbox{\textit{CAIE S2 2019 Q6 [10]}}