| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2021 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Multiple independent observations |
| Difficulty | Standard +0.3 This is a straightforward S2 question testing standard PDF techniques: integration for probability (part a with independence), expectation formula (part b), median equation setup (part c), and a simple interpretation question (part d). All parts follow textbook methods with no 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 integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X > 10) = \displaystyle\int_{10}^{20} \dfrac{3}{8000}(x-20)^2\, dx\) | M1 | Attempt integration of \(f(x)\), ignore limits. |
| \(= \left[\dfrac{3}{8000} \times \dfrac{(x-20)^3}{3}\right]_{10}^{20}\) or \(\dfrac{3}{8000}\left[\dfrac{x^3}{3} - \dfrac{40x^2}{2} + 400x\right]_{10}^{20} = \dfrac{1}{8000}\left[0-(-10)^3\right]\) | M1 | Substitute correct limits 10 to 20; or \(1 - \ldots\) limits 0 to 10 in *their* integral |
| \(\dfrac{1}{8}\) or \(0.125\) | A1 | SC Unsupported answer of \(\frac{1}{8}\) scores B1 only |
| \(\left(\dfrac{1}{8}\right)^2 = \dfrac{1}{64}\) or \(0.0156\) (3 sf) | B1 FT | FT *their* \(P(X > 10)\) dependent on first M1 gained |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\int_0^{20} \frac{3}{8000}(x^3 - 40x^2 + 400x)\,dx\) | M1 | Attempt integration of \(xf(x)\). Ignore limits. |
| \(\frac{3}{8000}\left[\frac{x^4}{4} - \frac{40x^3}{3} + \frac{400x^2}{2}\right]_0^{20}\) or \(\left[\frac{3x}{8000} \times \frac{(x-20)^3}{3}\right] - \frac{1}{8000}\left[\frac{(x-20)^4}{4}\right]\) | A1 | Correct integral (by expanding or by parts) |
| \(\frac{3}{8000}\left[\frac{160000}{4} - \frac{40 \times 8000}{3} + 200 \times 400\right]\) | M1 | Subst correct limits in their (4th degree) integral |
| \(5\) | A1 | |
| 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\int_0^m \frac{3}{8000}(x-20)^2\,dx = 0.5\) | M1 | Attempt to integrate \(f(x)\) and equate to 0.5. Ignore limits. |
| \(\left[\frac{3}{8000} \times \frac{(x-20)^3}{3}\right]_0^m = 0.5\) or \(\frac{3}{8000}\left[\frac{x^3}{3} - \frac{40x^2}{2} + 400x\right]_0^m = 0.5\) | M1 | Attempt integral and substitute limits \(0\) and \(m\), or \(m\) and \(20\) and \(= 0.5\) |
| \(\frac{1}{8000}\left[(m-20)^3 - (-20)^3\right] = 0.5\) | ||
| \((m-20)^3 = -4000\) | A1 | AG. Found convincingly. |
| \(m = 20 + \sqrt[3]{-4000}\), \(m = 4.13\) (3 sf) | B1 | |
| 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Doesn't allow for trains \(> 20\) mins late or Doesn't allow for trains being early | B1 | Or any relevant comment e.g. trains on Sun may be different to trains on Mon |
| 1 |
## Question 6:
### Part 6(a):
$P(X > 10) = \displaystyle\int_{10}^{20} \dfrac{3}{8000}(x-20)^2\, dx$ | **M1** | Attempt integration of $f(x)$, ignore limits.
$= \left[\dfrac{3}{8000} \times \dfrac{(x-20)^3}{3}\right]_{10}^{20}$ or $\dfrac{3}{8000}\left[\dfrac{x^3}{3} - \dfrac{40x^2}{2} + 400x\right]_{10}^{20} = \dfrac{1}{8000}\left[0-(-10)^3\right]$ | **M1** | Substitute correct limits 10 to 20; or $1 - \ldots$ limits 0 to 10 in *their* integral
$\dfrac{1}{8}$ or $0.125$ | **A1** | SC Unsupported answer of $\frac{1}{8}$ scores B1 only
$\left(\dfrac{1}{8}\right)^2 = \dfrac{1}{64}$ or $0.0156$ (3 sf) | **B1 FT** | FT *their* $P(X > 10)$ dependent on first M1 gained
## Question 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\int_0^{20} \frac{3}{8000}(x^3 - 40x^2 + 400x)\,dx$ | **M1** | Attempt integration of $xf(x)$. Ignore limits. |
| $\frac{3}{8000}\left[\frac{x^4}{4} - \frac{40x^3}{3} + \frac{400x^2}{2}\right]_0^{20}$ or $\left[\frac{3x}{8000} \times \frac{(x-20)^3}{3}\right] - \frac{1}{8000}\left[\frac{(x-20)^4}{4}\right]$ | **A1** | Correct integral (by expanding or by parts) |
| $\frac{3}{8000}\left[\frac{160000}{4} - \frac{40 \times 8000}{3} + 200 \times 400\right]$ | **M1** | Subst correct limits in their (4th degree) integral |
| $5$ | **A1** | |
| | **4** | |
---
## Question 6(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\int_0^m \frac{3}{8000}(x-20)^2\,dx = 0.5$ | **M1** | Attempt to integrate $f(x)$ and equate to 0.5. Ignore limits. |
| $\left[\frac{3}{8000} \times \frac{(x-20)^3}{3}\right]_0^m = 0.5$ or $\frac{3}{8000}\left[\frac{x^3}{3} - \frac{40x^2}{2} + 400x\right]_0^m = 0.5$ | **M1** | Attempt integral and substitute limits $0$ and $m$, or $m$ and $20$ and $= 0.5$ |
| $\frac{1}{8000}\left[(m-20)^3 - (-20)^3\right] = 0.5$ | | |
| $(m-20)^3 = -4000$ | **A1** | AG. Found convincingly. |
| $m = 20 + \sqrt[3]{-4000}$, $m = 4.13$ (3 sf) | **B1** | |
| | **4** | |
---
## Question 6(d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Doesn't allow for trains $> 20$ mins late **or** Doesn't allow for trains being early | **B1** | Or any relevant comment e.g. trains on Sun may be different to trains on Mon |
| | **1** | |6 Alethia models the length of time, in minutes, by which her train is late on any day by the random variable $X$ with probability density function given by
$$f ( x ) = \begin{cases} \frac { 3 } { 8000 } ( x - 20 ) ^ { 2 } & 0 \leqslant x \leqslant 20 \\ 0 & \text { otherwise } \end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item Find the probability that the train is more than 10 minutes late on each of two randomly chosen days.
\item Find $\mathrm { E } ( X )$.
\item The median of $X$ is denoted by $m$.
Show that $m$ satisfies the equation $( m - 20 ) ^ { 3 } = - 4000$, and hence find $m$ correct to 3 significant figures.
\item State one way in which Alethia's model may be unrealistic.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2021 Q6 [13]}}