| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2020 |
| 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 S2 probability density function question requiring standard techniques: integrating to find k, calculating E(X), and finding a percentile by solving F(x) = 0.6. All steps are routine applications of formulas with no conceptual challenges, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int_1^a \frac{k}{x^2}\,dx = 1\) | M1 | |
| \(k\left[-\frac{1}{x}\right]_1^a = 1\); \(k\left[1 - \frac{1}{a}\right] = 1\) | A1 | |
| \(k\left[\frac{a-1}{a}\right] = 1\); \(\left(k = \frac{a}{a-1}\right)\) | A1 | AG |
| Answer | Marks |
|---|---|
| \(\frac{a}{a-1}\int_1^a \frac{1}{x}\,dx\) | M1 |
| \(\frac{a}{a-1}\left[\ln x\right]_1^a\) | A1 |
| \(\frac{a\ln a}{a-1}\) | A1 |
| Answer | Marks |
|---|---|
| \(\frac{a}{a-1}\int_1^m \frac{1}{x^2}\,dx = \frac{3}{5}\) | M1 |
| \(\frac{a}{a-1}\left[-\frac{1}{x}\right]_1^m = \frac{3}{5}\); \(\frac{a}{a-1}\left[1 - \frac{1}{m}\right] = \frac{3}{5}\) | A1 |
| \(\frac{1}{m} = 1 - \frac{3(a-1)}{5a}\) or \(\frac{1}{m} = \frac{2a+3}{5a}\) | A1 |
| \(m = \frac{5a}{2a+3}\) | A1 |
## Question 6:
**Part 6(a):**
| $\int_1^a \frac{k}{x^2}\,dx = 1$ | M1 | |
| $k\left[-\frac{1}{x}\right]_1^a = 1$; $k\left[1 - \frac{1}{a}\right] = 1$ | A1 | |
| $k\left[\frac{a-1}{a}\right] = 1$; $\left(k = \frac{a}{a-1}\right)$ | A1 | AG |
**Part 6(b):**
| $\frac{a}{a-1}\int_1^a \frac{1}{x}\,dx$ | M1 | |
| $\frac{a}{a-1}\left[\ln x\right]_1^a$ | A1 | |
| $\frac{a\ln a}{a-1}$ | A1 | |
**Part 6(c):**
| $\frac{a}{a-1}\int_1^m \frac{1}{x^2}\,dx = \frac{3}{5}$ | M1 | |
| $\frac{a}{a-1}\left[-\frac{1}{x}\right]_1^m = \frac{3}{5}$; $\frac{a}{a-1}\left[1 - \frac{1}{m}\right] = \frac{3}{5}$ | A1 | |
| $\frac{1}{m} = 1 - \frac{3(a-1)}{5a}$ or $\frac{1}{m} = \frac{2a+3}{5a}$ | A1 | |
| $m = \frac{5a}{2a+3}$ | A1 | |6 A random variable $X$ has probability density function given by
$$f ( x ) = \begin{cases} \frac { k } { x ^ { 2 } } & 1 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$
where $k$ and $a$ are positive constants.
\begin{enumerate}[label=(\alph*)]
\item Show that $k = \frac { a } { a - 1 }$.
\item Find $\mathrm { E } ( X )$ in terms of $a$.
\item Find the 60th percentile of $X$ in terms of $a$.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2020 Q6 [10]}}