| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2012 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Find parameter from expectation |
| Difficulty | Moderate -0.3 This is a straightforward S2 question requiring standard techniques: verifying ∫f(x)dx=1 (routine integration), finding a parameter from E(X) using ∫xf(x)dx (direct calculation), and computing a probability P(X>6) (simple integration). All steps are mechanical applications of formulas with no conceptual challenges or novel problem-solving 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 integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f(x) \geq 0\) for all \(x\) defined | B1 | |
| \(\displaystyle\int_0^a \dfrac{2}{a^2} x\, dx\) | M1 | Attempt \(\int f(x)\,dx\) with limits \(0, a\). Must be \(a\) |
| \(= \left[\dfrac{2x^2}{2a^2}\right]_0^a = 1\) | A1 [3] | Or equivalent methods (e.g. by areas) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\displaystyle\int_0^a \dfrac{2}{a^2} x^2\, dx = 8\) | M1 | Attempt \(\int xf(x)\,dx\), ignore limits |
| \(\dfrac{2}{a^2}\left[\dfrac{x^3}{3}\right]_0^a = 8\) | A1 | Correct integrand and limits |
| \(\dfrac{2a}{3} = 8\) | A1 | |
| \(a = 12\) | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(1 - \displaystyle\int_0^6 \dfrac{2}{144} x\, dx\) or \(\displaystyle\int_6^{12} \dfrac{2}{144} x\, dx\) | M1 | Correct expression including limits; ft their \(a\) |
| \(= 1 - \dfrac{1}{72}\left[\dfrac{x^2}{2}\right]_0^6\) or \(\dfrac{1}{72}\left[\dfrac{x^2}{2}\right]_6^{12}\) | A1ft | Correct integrand and limits; ft their \(a\) |
| \(= \dfrac{3}{4}\) | A1ft [3] | ft their \(a\), dep \(0 < \text{ans} < 1\) |
## Question 6:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f(x) \geq 0$ for all $x$ defined | B1 | |
| $\displaystyle\int_0^a \dfrac{2}{a^2} x\, dx$ | M1 | Attempt $\int f(x)\,dx$ with limits $0, a$. Must be $a$ |
| $= \left[\dfrac{2x^2}{2a^2}\right]_0^a = 1$ | A1 [3] | Or equivalent methods (e.g. by areas) |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\displaystyle\int_0^a \dfrac{2}{a^2} x^2\, dx = 8$ | M1 | Attempt $\int xf(x)\,dx$, ignore limits |
| $\dfrac{2}{a^2}\left[\dfrac{x^3}{3}\right]_0^a = 8$ | A1 | Correct integrand and limits |
| $\dfrac{2a}{3} = 8$ | A1 | |
| $a = 12$ | [3] | |
### Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $1 - \displaystyle\int_0^6 \dfrac{2}{144} x\, dx$ or $\displaystyle\int_6^{12} \dfrac{2}{144} x\, dx$ | M1 | Correct expression including limits; ft their $a$ |
| $= 1 - \dfrac{1}{72}\left[\dfrac{x^2}{2}\right]_0^6$ or $\dfrac{1}{72}\left[\dfrac{x^2}{2}\right]_6^{12}$ | A1ft | Correct integrand and limits; ft their $a$ |
| $= \dfrac{3}{4}$ | A1ft [3] | ft their $a$, dep $0 < \text{ans} < 1$ |6 Darts are thrown at random at a circular board. The darts hit the board at distances $X$ centimetres from the centre, where $X$ is a random variable with probability density function given by
$$f ( x ) = \begin{cases} \frac { 2 } { a ^ { 2 } } x & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$
where $a$ is a positive constant.\\
(i) Verify that f is a probability density function whatever the value of $a$.
It is now given that $\mathrm { E } ( X ) = 8$.\\
(ii) Find the value of $a$.\\
(iii) Find the probability that a dart lands more than 6 cm from the centre of the board.
\hfill \mbox{\textit{CAIE S2 2012 Q6 [9]}}