| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2017 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Find median or percentiles |
| Difficulty | Moderate -0.3 This is a straightforward S2 question requiring standard techniques: integration to find E(X) and solving F(x)=0.5 for the median. The pdf is a simple linear function making both integrations routine. While it requires careful algebraic manipulation, it involves no conceptual difficulty beyond applying learned formulas, making it slightly easier than average. |
| Spec | 5.03c Calculate mean/variance: by integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{1}{4}\int_0^2 (x^2 + x)\,dx\) \(\left(= \frac{1}{4}\left[\frac{x^3}{3} + \frac{x^2}{2}\right]_0^2\right)\) | M1 | Attempt integ \(xf(x)\), ignore limits |
| \(= \frac{1}{4}\left(\frac{8}{3} + 2\right)\ (-0)\) | A1 | Subst correct limits in correct integration |
| \(= \frac{7}{6}\) OE or \(1.17\) (3 sf) | A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{1}{4}\int_0^m (x+1)\,dx = 0.5\) \(\left(=\frac{1}{4}\left[\frac{x^2}{2}+x\right]_0^m = 0.5\right)\) | M1 | attempt integ \(f(x)\), limits \(0\) to unknown (or unknown to \(2\)) and \(= 0.5\) |
| \(\frac{1}{4}\left(\frac{m^2}{2}+m\right) = 0.5\), \(m^2+2m-4=0\), \(m = \frac{-2 \pm \sqrt{4+16}}{2}\) OE | A1 | a correct equation in \(m\) (any form); or \(\sqrt{5}-1\) |
| \(m = 1.24\) | A1 | must reject the negative value if there |
| Total: 3 |
## Question 5(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{1}{4}\int_0^2 (x^2 + x)\,dx$ $\left(= \frac{1}{4}\left[\frac{x^3}{3} + \frac{x^2}{2}\right]_0^2\right)$ | M1 | Attempt integ $xf(x)$, ignore limits |
| $= \frac{1}{4}\left(\frac{8}{3} + 2\right)\ (-0)$ | A1 | Subst correct limits in correct integration |
| $= \frac{7}{6}$ OE or $1.17$ (3 sf) | A1 | |
| **Total: 3** | | |
## Question 5(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{1}{4}\int_0^m (x+1)\,dx = 0.5$ $\left(=\frac{1}{4}\left[\frac{x^2}{2}+x\right]_0^m = 0.5\right)$ | M1 | attempt integ $f(x)$, limits $0$ to unknown (or unknown to $2$) and $= 0.5$ |
| $\frac{1}{4}\left(\frac{m^2}{2}+m\right) = 0.5$, $m^2+2m-4=0$, $m = \frac{-2 \pm \sqrt{4+16}}{2}$ OE | A1 | a correct equation in $m$ (any form); or $\sqrt{5}-1$ |
| $m = 1.24$ | A1 | must reject the negative value if there |
| **Total: 3** | | |
---5 A continuous random variable, $X$, has probability density function given by
$$f ( x ) = \begin{cases} \frac { 1 } { 4 } ( x + 1 ) & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$
(i) Find $\mathrm { E } ( X )$.\\
................................................................................................................................. .\\
(ii) Find the median of $X$.\\
\hfill \mbox{\textit{CAIE S2 2017 Q5 [6]}}