| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2010 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Find expectation E(X) |
| Difficulty | Moderate -0.8 This is a straightforward S2 question requiring standard applications of formulas: E(X) uses ∫xf(x)dx, median solves ∫f(x)dx=0.5, and part (iii) applies independence with P(X>3)². All three parts are routine calculations with a simple linear pdf requiring only basic integration of polynomials. No problem-solving insight needed, just formula recall and execution. |
| 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 |
| \(\int_2^4 \dfrac{x^2}{6}\, dx \left(= \left[\dfrac{x^3}{18}\right]_2^4\right)\) | M1 | Attempt integ \(xf(x)\), ignore limits |
| \(= \dfrac{4^3}{18} - \dfrac{2^3}{18}\) | M1 | Subst correct limits in \(\dfrac{x^3}{n}\) |
| \(= \dfrac{28}{9}\) | A1 [3] | oe |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int_2^m \dfrac{x}{6}\, dx = \left[\dfrac{x^2}{12}\right]_2^m\) or \(\int_m^4 \dfrac{x}{6}\, dx\) | M1 | Attempt integ \(f(x)\) and \(= 0.5\) (ignore limits) |
| \(\dfrac{m^2}{12} - \dfrac{2^2}{12} = 0.5\) or \(\dfrac{4^2}{12} - \dfrac{m^2}{12} = 0.5\) | M1 | Attempt integ \(f(x)\), limits 2 to unknown or unknown to 4, or by areas |
| \(m = \sqrt{10}\) oe | A1 [3] | \(\sqrt{10}\) or 3.16 (3 sfs) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int_3^4 \dfrac{x}{6}\, dx = \left[\dfrac{x^2}{12}\right]_3^4 = \dfrac{7}{12}\) | M1* | Attempt integ \(f(x)\), one limit must be 3 |
| \(\left(\dfrac{7}{12}\right)^2\) | M1*dep | Square their \(\dfrac{7}{12}\) |
| \(= \dfrac{49}{144}\) or 0.340 (3 sfs) | A1 [3] |
## Question 5:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int_2^4 \dfrac{x^2}{6}\, dx \left(= \left[\dfrac{x^3}{18}\right]_2^4\right)$ | M1 | Attempt integ $xf(x)$, ignore limits |
| $= \dfrac{4^3}{18} - \dfrac{2^3}{18}$ | M1 | Subst correct limits in $\dfrac{x^3}{n}$ |
| $= \dfrac{28}{9}$ | A1 [3] | oe |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int_2^m \dfrac{x}{6}\, dx = \left[\dfrac{x^2}{12}\right]_2^m$ or $\int_m^4 \dfrac{x}{6}\, dx$ | M1 | Attempt integ $f(x)$ and $= 0.5$ (ignore limits) |
| $\dfrac{m^2}{12} - \dfrac{2^2}{12} = 0.5$ or $\dfrac{4^2}{12} - \dfrac{m^2}{12} = 0.5$ | M1 | Attempt integ $f(x)$, limits 2 to unknown or unknown to 4, or by areas |
| $m = \sqrt{10}$ oe | A1 [3] | $\sqrt{10}$ or 3.16 (3 sfs) |
### Part (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int_3^4 \dfrac{x}{6}\, dx = \left[\dfrac{x^2}{12}\right]_3^4 = \dfrac{7}{12}$ | M1* | Attempt integ $f(x)$, one limit must be 3 |
| $\left(\dfrac{7}{12}\right)^2$ | M1*dep | Square their $\dfrac{7}{12}$ |
| $= \dfrac{49}{144}$ or 0.340 (3 sfs) | A1 [3] | |
---5 A continuous random variable $X$ has probability density function given by
$$f ( x ) = \begin{cases} \frac { 1 } { 6 } x & 2 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$
(i) Find $\mathrm { E } ( X )$.\\
(ii) Find the median of $X$.\\
(iii) Two independent values of $X$ are chosen at random. Find the probability that both these values are greater than 3 .
\hfill \mbox{\textit{CAIE S2 2010 Q5 [9]}}