| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2017 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Parameter interpretation in context |
| Difficulty | Moderate -0.3 This is a straightforward continuous probability distribution question requiring standard techniques: recognizing that area under pdf relates to probability, integrating to find k, and computing E(X) and Var(X) using standard formulas. Part (i) requires only visual interpretation of a decreasing function, while parts (ii)-(iv) are routine A-level integration exercises with no novel problem-solving required. Slightly easier than average due to the simple rational function form. |
| 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 | Marks | Guidance |
| Greater area where \(x < 7.5\) than \(x > 7.5\) | B1 | Allow graph higher for \(x < 7.5\) than for \(x > 7.5\), or graph decreasing or equiv expl'n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\int_{5}^{10} \frac{k}{x^2}\, dx = 1\) | M1 | Attempt Integ \(f(x) = 1\) ignore limits |
| \(k\left[-\frac{1}{x}\right]_{5}^{10} = 1\), \(\quad k \times \frac{1}{10} = 1\) | A1 | Correct integration and limits |
| \(k = 10\) AG | A1 | No errors seen |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(10\int_{5}^{10} \frac{1}{x}\, dx\) | M1 | Attempt Integ \(xf(x)\) ignore limits |
| \(= 10\left[\ln x\right]_{5}^{10} = 10(\ln 10 - \ln 5)\) | M1 | Correct integration and limits |
| \(= 10\ln 2\) or \(6.93\) (3 sf) | A1 | OE |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(10\int_{5}^{10} 1\, dx - "6.93"^2\) | M1 | Attempt \((\text{Integ } x^2f(x)) - (E(x))^2\). No limits M0 |
| \(= 1.95\) (accept \(1.96\)) | A1 | *Use of 6.93 gives 1.97* A0 |
## Question 4(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Greater area where $x < 7.5$ than $x > 7.5$ | B1 | Allow graph higher for $x < 7.5$ than for $x > 7.5$, or graph decreasing or equiv expl'n |
## Question 4(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\int_{5}^{10} \frac{k}{x^2}\, dx = 1$ | M1 | Attempt Integ $f(x) = 1$ ignore limits |
| $k\left[-\frac{1}{x}\right]_{5}^{10} = 1$, $\quad k \times \frac{1}{10} = 1$ | A1 | Correct integration and limits |
| $k = 10$ AG | A1 | No errors seen |
## Question 4(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $10\int_{5}^{10} \frac{1}{x}\, dx$ | M1 | Attempt Integ $xf(x)$ ignore limits |
| $= 10\left[\ln x\right]_{5}^{10} = 10(\ln 10 - \ln 5)$ | M1 | Correct integration and limits |
| $= 10\ln 2$ or $6.93$ (3 sf) | A1 | OE |
## Question 4(iv):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $10\int_{5}^{10} 1\, dx - "6.93"^2$ | M1 | Attempt $(\text{Integ } x^2f(x)) - (E(x))^2$. No limits **M0** |
| $= 1.95$ (accept $1.96$) | A1 | *Use of 6.93 gives 1.97* **A0** |4\\
\includegraphics[max width=\textwidth, alt={}, center]{332f0909-c192-40f7-88b7-7cfec2db2eef-06_428_773_260_685}
The time, $X$ minutes, taken by a large number of runners to complete a certain race has probability density function f given by
$$f ( x ) = \begin{cases} \frac { k } { x ^ { 2 } } & 5 \leqslant x \leqslant 10 \\ 0 & \text { otherwise } \end{cases}$$
where $k$ is a constant, as shown in the diagram.\\
(i) Without calculation, explain how you can tell that there were more runners whose times were below 7.5 minutes than above 7.5 minutes.\\
(ii) Show that $k = 10$.\\
(iii) Find $\mathrm { E } ( X )$.\\
(iv) Find $\operatorname { Var } ( X )$.\\
\hfill \mbox{\textit{CAIE S2 2017 Q4 [9]}}