| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2014 |
| 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 | Moderate -0.3 This is a straightforward continuous probability distribution question requiring standard techniques: using the integral condition for part (i), calculating E(X) by integration for part (ii), and solving for the median using the cumulative distribution function for part (iii). All three parts follow routine procedures with no conceptual challenges beyond basic S2 content, 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 integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int_1^a \frac{k}{x}dx = 1\) | M1 | Int \(f(x)\) & equate to 1. Ignore limits |
| \(k[\ln x]_1^a = 1\) | A1 | Correct integration and limits and \(= 1\) |
| \(k\ln a = 1 \Rightarrow k = \frac{1}{\ln a}\) | A1 [3] | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{1}{\ln a}\int_1^a 1\,dx\) or \(k\int_1^a 1\,dx\) | M1 | Int \(xf(x)\). Ignore limits |
| \(= \frac{1}{\ln a}[x]_1^a\) or \(k[x]_1^a\) | A1 | Correct integration and limits (condone missing \(k\)) |
| \(= \frac{1}{\ln a}(a-1)\) | A1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{1}{\ln a}\int_1^m \frac{1}{x}dx = 0.5\) | M1 | Int \(f(x)\) and equate to 0.5. Ignore limits |
| \(\frac{1}{\ln a}[\ln x]_1^m = 0.5\) | A1 | Correct integration and limits (1 to \(m\) or \(m\) to \(a\)) (condone missing \(k\)) |
| \(\frac{1}{\ln a}\ln m = 0.5\) | ||
| \(\ln m = 0.5\ln a\) | A1 | or \(\ln m = \ln a^{0.5}\) |
| \(m = \sqrt{a}\) | A1 [4] |
## Question 7:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int_1^a \frac{k}{x}dx = 1$ | M1 | Int $f(x)$ & equate to 1. Ignore limits |
| $k[\ln x]_1^a = 1$ | A1 | Correct integration and limits and $= 1$ |
| $k\ln a = 1 \Rightarrow k = \frac{1}{\ln a}$ | A1 [3] | AG |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{1}{\ln a}\int_1^a 1\,dx$ or $k\int_1^a 1\,dx$ | M1 | Int $xf(x)$. Ignore limits |
| $= \frac{1}{\ln a}[x]_1^a$ or $k[x]_1^a$ | A1 | Correct integration and limits (condone missing $k$) |
| $= \frac{1}{\ln a}(a-1)$ | A1 [3] | |
### Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{1}{\ln a}\int_1^m \frac{1}{x}dx = 0.5$ | M1 | Int $f(x)$ and equate to 0.5. Ignore limits |
| $\frac{1}{\ln a}[\ln x]_1^m = 0.5$ | A1 | Correct integration and limits (1 to $m$ or $m$ to $a$) (condone missing $k$) |
| $\frac{1}{\ln a}\ln m = 0.5$ | | |
| $\ln m = 0.5\ln a$ | A1 | or $\ln m = \ln a^{0.5}$ |
| $m = \sqrt{a}$ | A1 [4] | |
---7 A random variable $X$ has probability density function given by
$$f ( x ) = \begin{cases} \frac { k } { x } & 1 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$
where $k$ and $a$ are positive constants.\\
(i) Show that $k = \frac { 1 } { \ln a }$.\\
(ii) Find $\mathrm { E } ( X )$ in terms of $a$.\\
(iii) Find the median of $X$ in terms of $a$.
\hfill \mbox{\textit{CAIE S2 2014 Q7 [10]}}