| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2012 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Single-piece PDF with k |
| Difficulty | Standard +0.3 This is a straightforward continuous probability distribution question requiring standard techniques: integrating to find k, solving for the median using cumulative distribution, and computing E(X). The integration of sin x is routine, and all three parts follow predictable methods with no novel problem-solving required. Slightly above average difficulty only due to the multi-part nature and need for numerical solving in part (ii). |
| 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 |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(k\int_0^{\frac{2\pi}{3}} \sin x \, dx = 1\) | M1 | Integ \(k = 1\), ignore limits |
| \(k[-\cos x]_0^{\frac{2\pi}{3}}\) | ||
| \(k\left[-\cos\frac{2\pi}{3} + \cos 0\right] = 1\) | Must see this line or next | |
| \(k[0.5 + 1] = 1\) | ||
| \(k = \frac{2}{3}\) (AG) | A1 (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{2}{3}\int_0^m \sin x \, dx = 0.5\) | M1* | Integ \(k = 0.5\), ignore limits |
| \(\frac{2}{3}[-\cos x]_0^m = 0.5\) | M1* dep | Correct integrand & limits 0 to unknown, \(= 0.5\) |
| \(\frac{2}{3}(-\cos m + 1) = 0.5\) | ||
| \(\cos m = 0.25\) | A1 | But allow \(a\cos m = b\) where \(b/a = 0.25\); dep \(\cos m = 0.25\) seen |
| \(m = 1.32\) (3 sf) (AG) | A1 (4) | NB accept full verification |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{2}{3}\int_0^{\frac{2\pi}{3}} x\sin x \, dx\) | M1 | Integ \(xf(x)\), ignore limits |
| \(= \frac{2}{3}\left\{[x(-\cos x)]_0^{\frac{2\pi}{3}} - \int_0^{\frac{2\pi}{3}}(-\cos x)\,dx\right\}\) | M1* dep | 1st step attempted i.e. \(x(-\cos x)\), ignore limits |
| \(= \frac{2}{3}\left\{\frac{\pi}{3} - 0 - [-\sin x]_0^{\frac{2\pi}{3}}\right\}\) | M1* dep | 2nd step attempted including correct limits applied |
| \(= \frac{2}{3}\left(\frac{\pi}{3} + \sin\frac{2\pi}{3}\right)\) | ||
| \(= \frac{2\pi + 3\sqrt{3}}{9}\) or \(1.28\) (3 sf) | A1 (4) | oe |
## Question 7:
### Part (i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $k\int_0^{\frac{2\pi}{3}} \sin x \, dx = 1$ | M1 | Integ $k = 1$, ignore limits |
| $k[-\cos x]_0^{\frac{2\pi}{3}}$ | | |
| $k\left[-\cos\frac{2\pi}{3} + \cos 0\right] = 1$ | | Must see this line or next |
| $k[0.5 + 1] = 1$ | | |
| $k = \frac{2}{3}$ (AG) | A1 (2) | |
### Part (ii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{2}{3}\int_0^m \sin x \, dx = 0.5$ | M1* | Integ $k = 0.5$, ignore limits |
| $\frac{2}{3}[-\cos x]_0^m = 0.5$ | M1* dep | Correct integrand & limits 0 to unknown, $= 0.5$ |
| $\frac{2}{3}(-\cos m + 1) = 0.5$ | | |
| $\cos m = 0.25$ | A1 | But allow $a\cos m = b$ where $b/a = 0.25$; dep $\cos m = 0.25$ seen |
| $m = 1.32$ (3 sf) (AG) | A1 (4) | NB accept full verification |
### Part (iii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{2}{3}\int_0^{\frac{2\pi}{3}} x\sin x \, dx$ | M1 | Integ $xf(x)$, ignore limits |
| $= \frac{2}{3}\left\{[x(-\cos x)]_0^{\frac{2\pi}{3}} - \int_0^{\frac{2\pi}{3}}(-\cos x)\,dx\right\}$ | M1* dep | 1st step attempted i.e. $x(-\cos x)$, ignore limits |
| $= \frac{2}{3}\left\{\frac{\pi}{3} - 0 - [-\sin x]_0^{\frac{2\pi}{3}}\right\}$ | M1* dep | 2nd step attempted including correct limits applied |
| $= \frac{2}{3}\left(\frac{\pi}{3} + \sin\frac{2\pi}{3}\right)$ | | |
| $= \frac{2\pi + 3\sqrt{3}}{9}$ or $1.28$ (3 sf) | A1 (4) | oe |7\\
\includegraphics[max width=\textwidth, alt={}, center]{7333c047-edad-4385-b3f8-248e8725cfcb-3_412_718_1037_715}
A random variable $X$ has probability density function given by
$$f ( x ) = \begin{cases} k \sin x & 0 \leqslant x \leqslant \frac { 2 } { 3 } \pi \\ 0 & \text { otherwise } \end{cases}$$
where $k$ is a constant, as shown in the diagram.\\
(i) Show that $k = \frac { 2 } { 3 }$.\\
(ii) Show that the median of $X$ is 1.32 , correct to 3 significant figures.\\
(iii) Find $\mathrm { E } ( X )$.
\hfill \mbox{\textit{CAIE S2 2012 Q7 [10]}}