| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2014 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Geometric/graphical PDF with k |
| Difficulty | Moderate -0.3 This is a straightforward PDF question requiring: (i) using the total area = 1 property to find c (simple geometry of trapezoid/triangle), (ii) setting up and solving an area equation for a given probability, and (iii) computing E(X) using standard integration. All steps are routine applications of S2 techniques with no novel insight required, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| Part (i) | ||
| \(\frac{1}{2}c^2 = 1\) | M1 | Area of triangle = 1 or integral of \(kx\) with limits 0 and \(c\) and equated to 1 |
| \(c = \sqrt{2}\) or 1.41 (3 sf) | A1 | [2] |
| Part (ii) | ||
| \(f(x) = x\) or \(y = x\) | B1 | Seen or implied, e.g. by next line. Can be awarded anywhere in the question. Implied by \((a+1)\) in area of trapezium. Ignore limits. Must be integral of \(kx\) and equated to 0.1. Or trapezium area. |
| \(\int_a^c xdx = 0.1\) | M1 | |
| \(\left[\frac{x^2}{2}\right]_a^c = 0.1\) | A1\(\checkmark\) | Correct limits, ft incorrect \(kx\). |
| \(1 - a^2 = 0.2\) \(a = 0.894\) (3 sf) | A1 | [4] |
| Part (iii) | ||
| \(\int_0^{\sqrt{2}} x^2 dx\) | M1 | Ignore limits; ft their \(f(x)\) but not \(\int xdx\) |
| \(\left[\frac{x^3}{3}\right]_0^{\sqrt{2}}\) | ||
| \(= \frac{2}{3}\sqrt{2}\) or 0.943 or \(\sqrt{\left(\frac{8}{3}\right)}\) | A1\(\checkmark\) | [2] ft their \(c\), dep \(0 < \) ans \(<\) their \(c\). Not ft their \(f(x)\) |
| Total | [8] |
| **Part (i)** |
|---|
| $\frac{1}{2}c^2 = 1$ | M1 | Area of triangle = 1 or integral of $kx$ with limits 0 and $c$ and equated to 1 |
| $c = \sqrt{2}$ or 1.41 (3 sf) | A1 | [2] |
| **Part (ii)** |
|---|
| $f(x) = x$ or $y = x$ | B1 | Seen or implied, e.g. by next line. Can be awarded anywhere in the question. Implied by $(a+1)$ in area of trapezium. Ignore limits. Must be integral of $kx$ and equated to 0.1. Or trapezium area. |
| $\int_a^c xdx = 0.1$ | M1 | |
| $\left[\frac{x^2}{2}\right]_a^c = 0.1$ | A1$\checkmark$ | Correct limits, ft incorrect $kx$. |
| $1 - a^2 = 0.2$ $a = 0.894$ (3 sf) | A1 | [4] |
| **Part (iii)** |
|---|
| $\int_0^{\sqrt{2}} x^2 dx$ | M1 | Ignore limits; ft their $f(x)$ but not $\int xdx$ |
| $\left[\frac{x^3}{3}\right]_0^{\sqrt{2}}$ | | |
| $= \frac{2}{3}\sqrt{2}$ or 0.943 or $\sqrt{\left(\frac{8}{3}\right)}$ | A1$\checkmark$ | [2] ft their $c$, dep $0 < $ ans $<$ their $c$. Not ft their $f(x)$ |
| **Total** | [8] |2\\
\includegraphics[max width=\textwidth, alt={}, center]{323cf83a-e23b-494e-a911-856d8f1c92fd-2_483_791_708_676}
The diagram shows the graph of the probability density function, f , of a random variable $X$.\\
(i) Find the value of the constant $c$.\\
(ii) Find the value of $a$ such that $\mathrm { P } ( a < X < 1 ) = 0.1$.\\
(iii) Find $\mathrm { E } ( X )$.
\hfill \mbox{\textit{CAIE S2 2014 Q2 [8]}}