| 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 | Compare mean and median using probability |
| Difficulty | Standard +0.3 This is a straightforward continuous probability distribution question requiring standard techniques: computing E(X) by integration, finding a probability by integration, and interpreting the relationship between mean and median. All steps are routine S2 material 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 integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{2}{3}\int_1^6 x^2\,dx\) | M1 | Attempt integrating \(xf(x)\); ignore limits |
| \(= \frac{2}{3}\left[\frac{x^3}{3}\right]_1^2\) | A1 | Correct integration and limits |
| \(= \frac{14}{9}\) or \(1.56\) o.e. | A1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{2}{3}\int_1^{^9\!4} x\,dx\) | M1 | Attempt integrating \(f(x)\); with limits |
| \(\left(=\frac{2}{3}\left[\frac{x^3}{3}\right]_1^2\right)\) | ||
| \(= \frac{115}{243}\) or \(0.473\) (3 s.f.) | A1 [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{115}{243} < \frac{1}{2}\) o.e. | M1 | Comparison of prob. or values |
| Hence mean \(<\) median | A1ft [2] | ft (i) or (ii) |
## Question 2:
### Part (i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{2}{3}\int_1^6 x^2\,dx$ | M1 | Attempt integrating $xf(x)$; ignore limits |
| $= \frac{2}{3}\left[\frac{x^3}{3}\right]_1^2$ | A1 | Correct integration and limits |
| $= \frac{14}{9}$ or $1.56$ o.e. | A1 [3] | |
### Part (ii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{2}{3}\int_1^{^9\!4} x\,dx$ | M1 | Attempt integrating $f(x)$; with limits |
| $\left(=\frac{2}{3}\left[\frac{x^3}{3}\right]_1^2\right)$ | | |
| $= \frac{115}{243}$ or $0.473$ (3 s.f.) | A1 [2] | |
### Part (iii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{115}{243} < \frac{1}{2}$ o.e. | M1 | Comparison of prob. or values |
| Hence mean $<$ median | A1ft [2] | ft (i) or (ii) |
---2 A random variable $X$ has probability density function given by
$$f ( x ) = \begin{cases} \frac { 2 } { 3 } x & 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$
(i) Find $\mathrm { E } ( X )$.\\
(ii) Find $\mathrm { P } ( X < \mathrm { E } ( X ) )$.\\
(iii) Hence explain whether the mean of $X$ is less than, equal to or greater than the median of $X$.
\hfill \mbox{\textit{CAIE S2 2013 Q2 [7]}}