| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Single-piece PDF with k |
| Difficulty | Moderate -0.8 This is a straightforward PDF question requiring standard techniques: integrating to find k, calculating a probability, and finding an expectation. All three parts follow routine procedures with simple integration of power functions. The only minor challenge is recognizing the improper integral converges, but this is typical for S2 level. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| (i) | \(\int_1^\infty \frac{k}{x^3}\,dx = 1\); \(\left[-\frac{k}{2x^2}\right]_1^\infty = 1\); \(0 - \left(-\frac{k}{2}\right) = 1\) | M1, A1 [2] |
| (ii) | \(\int_1^2 \frac{2}{x^3}\,dx\); \(= \left[-\frac{1}{x^2}\right]_1^2\); \(= \frac{3}{4}\) | M1, A1 [2] |
| (iii) | \(\int_1^\infty \frac{2}{x^2}\,dx\); \(= \left[-\frac{2}{x}\right]_1^\infty\); \(= 2\) | M1, A1, A1 [3] |
## Question 5:
**(i)** | $\int_1^\infty \frac{k}{x^3}\,dx = 1$; $\left[-\frac{k}{2x^2}\right]_1^\infty = 1$; $0 - \left(-\frac{k}{2}\right) = 1$ | M1, A1 [2] | All correct including limits and attempt to integrate; or $0 + \frac{k}{2} = 1$ or $\frac{k}{2} = 1$ AG must be convincing |
**(ii)** | $\int_1^2 \frac{2}{x^3}\,dx$; $= \left[-\frac{1}{x^2}\right]_1^2$; $= \frac{3}{4}$ | M1, A1 [2] | Attempt integ $f(x)$; ignore limits |
**(iii)** | $\int_1^\infty \frac{2}{x^2}\,dx$; $= \left[-\frac{2}{x}\right]_1^\infty$; $= 2$ | M1, A1, A1 [3] | Attempt integ $xf(x)$; ignore limits; correct & correct limits |
---5 A random variable $X$ has probability density function given by
$$\mathrm { f } ( x ) = \begin{cases} \frac { k } { x ^ { 3 } } & x \geqslant 1 \\ 0 & \text { otherwise } \end{cases}$$
where $k$ is a constant.\\
(i) Show that $k = 2$.\\
(ii) Find $\mathrm { P } ( 1 \leqslant X \leqslant 2 )$.\\
(iii) Find $\mathrm { E } ( X )$.
\hfill \mbox{\textit{CAIE S2 2013 Q5 [7]}}