| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2019 |
| Session | March |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Piecewise PDF with multiple regions |
| Difficulty | Standard +0.3 This is a straightforward S2 continuous probability distribution question requiring standard techniques: integration of a simple trigonometric PDF for probability, solving an equation for the median, and computing expectation. All three parts use routine calculus with cos x, which is simpler than most PDF questions. The single-region PDF and standard trigonometric integral make this easier than average A-level maths questions. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03c Calculate mean/variance: by integration5.03e Find cdf: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\sqrt{2}\int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \cos x \, dx = \sqrt{2}\left[\sin x\right]_{\frac{\pi}{6}}^{\frac{\pi}{4}}\) | M1 | Attempt integ \(f(x)\) with correct limits |
| \(= \frac{2-\sqrt{2}}{2}\) oe or \(0.293\) (3 sf) | A1 | SC Final answer of \(0.707\) scores B1sc |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\sqrt{2}\int_{0}^{m} \cos x \, dx = 0.5\) | M1 | Attempt to integ \(f(x)\) & \(= 0.5\). Ignore limits. Condone missing \(\sqrt{2}\) |
| \(\sqrt{2}\left[\sin x\right]_0^m = 0.5\), \(\sqrt{2}\sin m = 0.5\) | A1 | Correct integral and limits 0 to unknown & \(= 0.5\) Condone missing \(\sqrt{2}\) |
| \(\sin m = \frac{1}{2\sqrt{2}}\) oe | M1 | For rearranging their expression to the form \(\sin m =\ldots\) (\(\sin m = 0.35355\ldots\) or \(0.354\)) seen or implied |
| \(m = 0.361\) (3 sfs) | A1 | No errors seen (Note \(20.705\) can score M1 A1 M1 A0) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\sqrt{2}\int_0^{\frac{\pi}{4}} x\cos x\, dx\) | M1 | Attempt to integrate \(xf(x)\). Ignore limits. Condone missing \(\sqrt{2}\) |
| \(= \sqrt{2}\left\{[x(\sin x)]_0^{\frac{\pi}{4}} - \int_0^{\frac{\pi}{4}} \sin x\, dx\right\}\) | M1 | Attempt to integrate by parts leading to expression of form \(\pm x\sin x \pm \cos x\) with correct limits |
| \(= \sqrt{2}\left\{\frac{\pi}{4\sqrt{2}} - 0 - [-\cos x]_0^{\frac{\pi}{4}}\right\}\) | A1 | For \(\sqrt{2}(x\sin x -(-\cos x))\) with correct limits |
| \(= \sqrt{2}\left\{\frac{\pi}{4\sqrt{2}} + \cos\frac{\pi}{4} - 1\right\}\) | A1 | |
| \(= \frac{\pi}{4} + 1 - \sqrt{2}\) oe or \(0.371\) (3 sf) | ||
| Total: 4 marks |
## Question 7(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\sqrt{2}\int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \cos x \, dx = \sqrt{2}\left[\sin x\right]_{\frac{\pi}{6}}^{\frac{\pi}{4}}$ | M1 | Attempt integ $f(x)$ with correct limits |
| $= \frac{2-\sqrt{2}}{2}$ oe or $0.293$ (3 sf) | A1 | SC Final answer of $0.707$ scores B1sc |
---
## Question 7(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\sqrt{2}\int_{0}^{m} \cos x \, dx = 0.5$ | M1 | Attempt to integ $f(x)$ & $= 0.5$. Ignore limits. Condone missing $\sqrt{2}$ |
| $\sqrt{2}\left[\sin x\right]_0^m = 0.5$, $\sqrt{2}\sin m = 0.5$ | A1 | Correct integral and limits 0 to unknown & $= 0.5$ Condone missing $\sqrt{2}$ |
| $\sin m = \frac{1}{2\sqrt{2}}$ oe | M1 | For rearranging their expression to the form $\sin m =\ldots$ ($\sin m = 0.35355\ldots$ or $0.354$) seen or implied |
| $m = 0.361$ (3 sfs) | A1 | No errors seen (Note $20.705$ can score M1 A1 M1 A0) |
## Question 7(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\sqrt{2}\int_0^{\frac{\pi}{4}} x\cos x\, dx$ | **M1** | Attempt to integrate $xf(x)$. Ignore limits. Condone missing $\sqrt{2}$ |
| $= \sqrt{2}\left\{[x(\sin x)]_0^{\frac{\pi}{4}} - \int_0^{\frac{\pi}{4}} \sin x\, dx\right\}$ | **M1** | Attempt to integrate by parts leading to expression of form $\pm x\sin x \pm \cos x$ with correct limits |
| $= \sqrt{2}\left\{\frac{\pi}{4\sqrt{2}} - 0 - [-\cos x]_0^{\frac{\pi}{4}}\right\}$ | **A1** | For $\sqrt{2}(x\sin x -(-\cos x))$ with correct limits |
| $= \sqrt{2}\left\{\frac{\pi}{4\sqrt{2}} + \cos\frac{\pi}{4} - 1\right\}$ | **A1** | |
| $= \frac{\pi}{4} + 1 - \sqrt{2}$ oe or $0.371$ (3 sf) | | |
| **Total: 4 marks** | | |7\\
\includegraphics[max width=\textwidth, alt={}, center]{a93e5413-6ad8-4957-8efd-470cf79792e2-12_428_693_260_724}
A random variable $X$ has probability density function given by
$$f ( x ) = \begin{cases} ( \sqrt { } 2 ) \cos x & 0 \leqslant x \leqslant \frac { 1 } { 4 } \pi \\ 0 & \text { otherwise } \end{cases}$$
as shown in the diagram.\\
(i) Find $\mathrm { P } \left( X > \frac { 1 } { 6 } \pi \right)$.\\
(ii) Find the median of $X$.\\
(iii) Find $\mathrm { E } ( X )$.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.\\
\hfill \mbox{\textit{CAIE S2 2019 Q7 [10]}}